Integral Exponents

For any n a natural number and a any real number:

with the interpretation that a ⁰ = 1 if a ≠ 0 .

So, for example:
3 ⁴= 3 x 3 x 3 x 3 = 81

Negative Exponent

If a is any non-zero number and n is a positive number then,

provided a ≠ 0 .

Example 1

Example 2

Example 3

Product of Exponent

If a is any real number and m & n are integers,

Example 1

Example 2

Quotient of Exponent

If a is any non-zero number and m & n are integers,

Example 1

Example 2

Properties of Exponent

If a & b are any real and m & n are integers,

Example

Exercises

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Exercises

Exercise for Exponents

Simplify the following monomials, just click on the buttons to see them

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Rational Exponents

Rational Exponents is in the form

where both m and n are integers and b any real number (except b cannot be negative when n is even).

Examples

We will start simple by looking the following special case, b integer.

is equivalent to

In other words, when evaluating we are really asking what number (in this case a) did we raise to the n to get b. Often is called the nth root of b

Examples

For m and n natural numbers and bany real number (except b cannot be negative when n is even)

and

Examples

Radicals

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Radicals

If n is a positive integer that is greater than 1 and a is a real number then,

where n is called the index, a is called the radicand, and the symbol is called the radical.
The left side of this equation is often called the radical form and the right side is often called the exponent form.

Note as well that the index is required in these to make sure that we correctly evaluate the radical.
There is one exception to this rule and that is square root.

For square roots we have, In other words, for square roots just drop the index.

Examples

We can also write the general rational exponent in terms of radicals as follows:

Properties of Radicals

If n is a positive integer greater than 1 and both a and b are positive real numbers

Example 1

Example 2

Example

A radical is said to be in simplified radical form (or just simplified form) if each of the following are true:

All exponents in the radicand must be less than the index.
Any exponents in the radicand can have no factors in common with the index.
No fractions appear under a radical.
No radicals appear in the denominator of a fraction.

Example 1

Answer

Example 2

Answer

Example 3

Answer

Example 4

Answer

Example 5

Answer

Example 6

Answer

Example 7

Answer

The Process of getting rid of the radicands in the denominator is called rationalizing the denominator.

Example 1

Answer

Example 2

Answer

Example 3

Answer

Example 4

Answer

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Polynomials

An algebraic expression involving only the operations of addition, subtraction, multiplication and raising to whole number powers on variables and constants is called POLYNOMIALS.

Examples

In a polynomial, a variable cannot appear in a denominator, as an exponent, or within a radical.

Accordingly, a polynomials in one variable x is constructed by adding or subtracting constants and terms of the form axⁿ , where a is real number and n is a natural number.

A polynomial in two variables x and y is constructed by adding and subtracting constants and terms of the form ax^myⁿ, where a is a real number and, m and n are natural numbers. Polynomials in three or more variables are defined in similar manner.

Polynomials are classified according to their degree. The power of the variable is the degree of that term if a term in a polynomial has only one variable as a factor and the sum of the powers of the variables if two or more variables are present in a term as a factor.

The degree of polynomials is the degree of the nonzero term with the highest degree in the polynomials. Any nonzero constant is defined to be a polynomial of degree 0.

The number 0 is also polynomials but is not assigned a degree.

POLYNOMIALS and NONPOLYNOMIALS

Polynomials in one variable:

The degree of the first term is 3, the degree of the second term is 1, the degree of the third term is 0, and the degree of the whole polynomials is 3.
Polynomials in several variables:

The degree of the first term is 5, the degree of the second term is 6, and the degree of the whole polynomial is 6.
Non-Polynomials:

A monomial is a polynomial that consists of exactly one term. A binomial is a polynomial that consists of exactly two terms. Finally, a multinomial is a polynomial that consists of more than two terms.

Examples

Monomial
Binomial
Trinomial
Multinomial

Adding and Subtracting Polynomials

Adding and Subtracting Polynomials can be done by suppressing the parentheses and combining like terms.

Examples

MULTIPLYING POLYNOMIALS

Multiplying polynomials involve the extensive use of distributive properties for real numbers, as well as real number properties.

Examples

COMBINED OPERATIONS

Polynomial Operations using all the properties of real numbers and Properties of exponents.

Examples

Square Polynomials

Cube Polynomials

Factoring Polynomials

Factoring Quadratic Polynomials

Special Factorization

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Special Polynomials (Square)

SQUARE OF A BINOMIAL

For all real numbers a and b the following identity holds

Sum

Difference

Examples

DIFFERENCE OF SQUARES

For Real numbers a and b the following identity holds

Examples

Exercises

Use Special Product to square each binomial.

Find the product

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Sample Problems Measurements and Vectors

CONVERSION OF UNITS

1. 2 ½ WEEKS
2. ¼ miles
3. 5 yards
4. 250 lbs
5. 4500 ml

=___________hours
=___________meters
=___________feet
=___________kg
=___________liters

6. 350 cm
7. 1.5 years
8. 25 days
9. 2.5 Mhz
10. ¾ kg

=___________μm
=___________hours
=___________minutes
=___________hertz
=___________centigram

PROBLEM SOLVING Measurements

Who is taller? Mark, 5’4” or Russel, 162 cm? Why? (1 in = 2.54 cm)
Which is heavier? ---345 μg or 3.45 x 10^-7 g? Why?
A desktop is advertize to have clock cycle speed of 2800 MHz. What is its frequency in Ghz?
A cylindrical water tank can hold 500 liters of water determine its diameter if it is 2 meters tall.
Convert from scientific notation into normal notation or Vice Versa
1. 5.3 X 10^-3
2. 1.63 × 10^-5
3. 8.34 X 10⁵
4. 2.4 × 10⁷
5. 0.000002631
6. 7.577
7. 5.56 X 10⁷
8. 573,100,000,000
9. 3.15 x 10³
10. 4

RESOLVE INTO COMPONENTS Vectors

FINDING VECTOR

PROBLEM SOLVING Vectors

A car is driven 215 km west and then 85 km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
A delivery truck travels 18 blocks north, 10 blocks east, and 16 blocks south. What is its final displacement from the origin? Assume the blocks are equal length.
A man walks from point A to point B which is 12km NE. From point B the man walks a further 8km east to point C. Calculate the man's resultant displacement.
Analitically determine the resultant of the following vector displacements: (1) 34 m, 25º north of east; (2) 48 m, 33º east of north.
An airplane travels 100 miles to the north, six hundred miles to the southeast and then 200 miles to the east. Find its displacement (resultant) from the starting point.
Two teenagers push a heavy crate across the floor. Dion pushes with a force of 185 N at 0°. Shirley exerts a force of 165 N at 30°. What is the resultant force on the crate?
A motorboat travels at 8.5 m/s. It heads straight across a river 110 m wide. If the water flows downstream at a rate of 3.8 m/s, what is the boat’s resultant velocity?
A 42-km/h wind blows toward 215°, while a plane heads toward 125° at 152 km/h. What is the resultant velocity of the plane?
A heavy box is pulled across a wooden floor with a rope. The rope makes an angle of 60° with the floor. A force of 75 N is exerted on the rope. What is the component of the force parallel to the floor?
Two forces act on an object. One force is 6.0 N horizontally. The second force is 8.0 N vertically. Find the magnitude and direction of the resultant.
A 62-N force acts at 30° and a second 62-N force acts at 60°. Determine the resultant force.
An airplane travels 100 miles to the north, six hundred miles to the southeast and then 200 miles to the east. Find its displacement from the starting point.
Use component method to find the resultant of the following set of forces: a) 1200 N,EAST; b) 300 N,60°NORTH of EAST. c) 100 N @ SOUTHWEST; d) 200 N, south.

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Sample Problem on Kinematics

PROBLEM SOLVING Kinematics

What must be your car’s average speed in order to travel 235 km in 3.25 h?
A bird can fly 25 km/hr How long does it take to fly 15 km?
If you are driving 110 km/hr along a straight road and you look to the side for 2.0 s, how far do you travel during this inattentive period?
Convert 35 mi/hr to (a)km/hr (b)m/s and (c)ft/s
You are driving home from school steadily at 95 km/hr for 130 km. It then begins to rain and you slow to 65 km/hr. You arrive home after driving 3 hours and 20 minutes. (a) How far is your hometown from school? (b) What was your average speed?
A person jogs eight complete laps around a quarter-mile track in a total time of 12.5 min. Calculate (a) the average speed and (b) the average velocity, in m/s
A horse canters away from its trainer in a straight line, moving 116 m away in 14.0 s. It then turns abruptly and gallops halfway back in 4.8 s. Calculate (a) its average speed and (b) its average velocity for the entire trip, using “away from the trainer” as the positive direction.
A car traveling 20 m/s is 110 m behind a truck traveling 15 m/s. How long will it take the car to reach the truck?
An airplane travels 3100 km at a speed of 790 km/hr and then encounters a tailwind that boosts its speed to 990 km/hr for the next 2800 km. What was the total time for the trip? What was the average speed of the plane for this trip?
Calculate the average speed and average velocity of a complete round-trip in which the outgoing 250 km is covered at 95 km/hr followed by a 1.0-hour lunch break, and the return 250 km is covered at 55 km/hr.
A sports car accelerates from rest to 31 m/s in 6.2 s. What is its average acceleration in m/s²
A sprinter accelerates from rest to 10.0 m/s in 1.35 s. What is her acceleration in m/s²?
At highway speeds, a particular automobile is capable of an acceleration of about 1.6 m/s². At this rate, how long does it take to accelerate from 80 km/hr to 110 km/hr.
A sports car moving at constant speed travels 110 m in 5.0 s. If it then brakes and comes to a stop in 4.0 s, what is its acceleration in m/s²? Express the answer in terms of “g’s,” where 1.0 g =9.8 m/s².
A car accelerates from 13 m/s to 25 m/s in 6.0 s. What was its acceleration? How far did it travel in this time? Assume constant acceleration.
A car slows down from 23 m/s to rest in a distance of 85 m. What was its acceleration, assumed constant?
A light plane must reach a speed of 33 m/s for takeoff. How long a runway is needed if the (constant) acceleration is 3.0 m/s²?
A world-class sprinter can burst out of the blocks to essentially top speed (of about 11.5m/s) in the first 15.0 m of the race. What is the average acceleration of this sprinter, and how long does it take her to reach that speed?
A car slows down uniformly from a speed of 21.0 m/s to rest in 6.00 s. How far did it travel in that time?
In coming to a stop, a car leaves skid marks 92 m long on the highway. Assuming a deceleration of 7.0 m/s² estimate the speed of the car just before braking.
A car traveling 85 km/h strikes a tree. The front end of the car compresses and the driver comes to rest after traveling 0.80 m. What was the average acceleration of the driver during the collision? Express the answer in terms of “g’s,” where 1.0 g =9.8 m/s².
Determine the stopping distances for a car with an initial speed of 95 km/h and human reaction time of 1.0 s, for an acceleration (a) a = -4.0 m/s² (b) a = -8.0 m/s²

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Sample Problem Free-Falling Bodies

PROBLEM SOLVING Free-Falling Bodies

(I) A stone is dropped from the top of a cliff. It hits the ground below after 3.25 s. How high is the cliff?
(I) If a car rolls gently (v_o= 0) off a vertical cliff, how long does it take it to reach 85 km/h?
(I) Estimate (a) how long it took King Kong to fall straight down from the top of the Empire State Building (380 m high), and (b) his velocity just before “landing”?
(II) A baseball is hit nearly straight up into the air with a speed of 22 m/s (a) How high does it go? (b) How long is it in the air?
(II) A ballplayer catches a ball 3.0 s after throwing it vertically upward. With what speed did he throw it, and what height did it reach?
(II) If air resistance is neglected, show (algebraically) that a ball thrown vertically upward with a speed v_o will have the same speed, v_o , when it comes back down to the starting point.
(II) A stone is thrown vertically upward with a speed of 18.0 m/s. (a) How fast is it moving when it reaches a height of 11.0 m? (b) How long is required to reach this height? (c) Why are there two answers to (b)?
(III) A falling stone takes 0.28 s to travel past a window 2.2 m tall. From what height above the top of the window did the stone fall?
(II) A helicopter is ascending vertically with a speed of 5.20 m/s. At a height of 125 m above the Earth, a package is dropped from a window. How much time does it take for the package to reach the ground? [Hint: The package’s initial speed equals the helicopter’s.]
(III) A rock is dropped from a sea cliff, and the sound of it striking the ocean is heard 3.2 s later. If the speed of sound is 340 m/s how high is the cliff?
(III) Suppose you adjust your garden hose nozzle for a hard stream of water. You point the nozzle vertically upward at a height of 1.5 m above the ground. When you quickly move the nozzle away from the vertical, you hear the water striking the ground next to you for another 2.0 s. What is the water speed as it leaves the nozzle?
(III) A stone is thrown vertically upward with a speed of 12.0 m/s from the edge of a cliff 70.0 m high. (a) How much later does it reach the bottom of the cliff? (b) What is its speed just before hitting? (c) What total distance did it travel?

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Sample Problem Force

PROBLEM SOLVING Newton's Laws of Motion

What force is needed to accelerate a child on a sled (total mass = 60.0 kg) at 1.25 m/s²?
A net force of 265 N accelerates a bike and rider at 2.30 m/s². What is the mass of the bike and rider together?
How much tension must a rope withstand if it is used to accelerate a 960-kg car horizontally along a frictionless surface at 1.20 m/s²?
What is the weight of a 76-kg astronaut
(a) on Earth,
(b) on the Moon ( g = 1.7m/s²),
(c) on Mars ( g= 3.7m/s²),
(d) in outer space traveling with constant velocity?
5. A 20.0-kg box rests on a table.
(a) What is the weight of the box and the normal force acting on it?
(b) A 10.0-kg box is placed on top of the 20.0-kg box. Determine the normal force that the table exerts on the 20.0-kg box and the normal force that the 20.0-kg box exerts on the 10.0-kg box.
What average force is required to stop an 1100-kg car in 8.0 s if the car is traveling at 30 m/s?
What average force is needed to accelerate a 7.00-gram pellet from rest to 125 m/s over a distance of 0.800 m along the barrel of a rifle?
A 0.140-kg baseball traveling 35 m/s strikes the catcher’s mitt, which, in bringing the ball to rest, recoils backward 11.0 cm. What was the average force applied by the ball on the glove?
A particular race car can cover a quarter-mile track (402 m) in 6.40 s starting from a standstill. Assuming the acceleration is constant, how many “g’s” does the driver experience? If the combined mass of the driver and race car is 485 kg, what horizontal force must the road exert on the tires?
A 12.0-kg bucket is lowered vertically by a rope in which there is 163 N of tension at a given instant. What is the acceleration of the bucket? Is it up or down?
A person stands on a bathroom scale in a motionless elevator. When the elevator begins to move, the scale briefly reads only 0.75 of the person’s regular weight. Calculate the acceleration of the elevator, and find the direction of acceleration.
A car of mass m=1200 kg is traveling at a speed of 25 m/s. Suddenly the brakes are applied and the car is brought to a stop over a distance of 20 m. Assuming constant breaking force find:
a.) the magnitude of the breaking force,
b.)the time required to stop.
c.)What will be the stopping distance if the initial speed is 50m/s?
Two blocks, m₁ = 2.0 kg and m₂ = 3.0 kg are put in contact on a frictionless surface. A horizontal force F = 5.0 N is applied to one of them (see Figure below).
a.)Find the force F_a between the two blocks.
b.)Find the force F_b if the force F is applied to m₂in the opposite direction.

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Sample Problem on Friction

A desk has a mass of 2.5 kilograms. If the coefficient of static friction between the desk and the floor is 0.04, what force must be used to move the desk from rest?
Once the desk above is in motion, what force must be used to keep it moving at a constant velocity if the coefficient of kinetic friction is 0.0267?
A 945-kg car traveling rightward at 22.6 m/s slams on the brakes and skids to a stop (with locked wheels). If the coefficient of friction between tires and road is 0.972, determine the distance required to stop.
The coefficient of static friction between a person’s shoe and the ground is 0.70. Determine the maximum magnitude of acceleration of the 62 kg person, if he starts running on a horizontal surface from rest.
A 250 kg box slides down a ramp and then across a level floor. The coefficient of kinetic friction along the floor is 0.20. A person sees the box moving at 1.0 m/s [left] and pushes on it with a horizontal force of 140 N [right]. How far does the box travel before coming to rest?
A 0.170 kg hockey puck is initially moving at 21.2 m/s [W] along the ice. The coefficient of kinetic friction for the puck and the ice is 0.005.
(a) What is the speed of the puck after travelling 58.5 m?
(b) After being played on for a while, the ice becomes rougher and the coefficient of kinetic friction increases to 0.047. How far will the puck travel if its initial and final speeds are the same as before?
A snowmobile is used to pull two sleds across the ice. The mass of the snowmobile and the rider is 320 kg. The mass of the first sled behind the snowmobile is 120 kg and the mass of the second sled is 140 kg. The ground exerts a force of 1500 N [forward] on the snowmobile. The coefficient of kinetic friction for the sleds on ice is 0.15. Assume that no other frictional forces act on the snowmobile. Calculate the acceleration of the snowmobile and sleds.
8. A string is tied to a 3.2 kg object on a table and a 1.5 kg object hanging over a pulley. The coefficient of kinetic friction between the 3.2 kg object and the table is 0.30.
(a) Calculate the acceleration of each object.
(b) Determine the magnitude of the tension in the string.
(c) How far will the objects move in 1.2 s if the initial velocity of the 3.2 kg object is 1.3 m/s [right]?

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Sample Problem Tensions

PROBLEM SOLVING Tensions

A fisherman yanks a fish vertically out of the water with an acceleration of 2.5 m/s² using very light fishing line that has a breaking strength of 22 N. The fisherman unfortunately loses the fish as the line snaps. What can you say about the mass of the fish?
How much tension must a rope withstand if it is used to accelerate a 1200-kg car vertically upward at 0.80 m/s²?
A 12.0-kg bucket is lowered vertically by a rope in which there is 163 N of tension at a given instant. What is the acceleration of the bucket? Is it up or down?
An elevator (mass 4850 kg) is to be designed so that the maximum acceleration is 0.0680g. What are the maximum and minimum forces the motor should exert on the supporting cable?
A 75-kg petty thief wants to escape from a third-story jail window. Unfortunately, a makeshift rope made of sheets tied together can support a mass of only 58 kg. How might the thief use this “rope” to escape? Give a quantitative answer.
An elevator has a mass of 1400kg. What is the tension in the supporting cable when the elevator traveling down at 10 m/s is brought to rest in a distance of 40 m. Assume a constant acceleration.
An object is hung from a spring balance attached to the ceiling of an elevator. The balance reads F₁ = 12 N when the elevator is accelerating upward, and reads F₂ = 8 N when it is accelerating downward with acceleration of the same magnitude a. Find the mass of this object and magnitude of acceleration a. We assume that gravitational acceleration g is known.
A man with a mass of m₁=80kg lowers himself by h=10m along a wall, while fixed to a rope that runs over a frictionless pulley to a m₂=70kg sandbag.
(a) What is his final speed if he started from a state of rest?
(b) How long does it take to “travel” this distance?
A lamp hangs vertically from a cord in an elevator which is descending with an downward acceleration of a=2.0m/s². The tension in the cord is T=10.0N. What is the mass m of this lamp?

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Sample Problem Work and Energy

PROBLEM SOLVING Work

(I) How much work is done by the gravitational force when a 265-kg pile driver falls 2.80 m?
(I) A 65.0-kg firefighter climbs a flight of stairs 20.0 m high. How much work is required?
(I) A 1300-N crate rests on the floor. How much work is required to move it at constant speed (a) 4.0 m along the floor against a friction force of 230 N, and (b) 4.0 m vertically?
(I) How much work did the movers do (horizontally) pushing a 160-kg crate 10.3 m across a rough floor without acceleration, if the effective coefficient of friction was 0.50?
(II) A box of mass 5.0 kg is accelerated by a force across a floor at a rate of 2.0 m/s² for 7.0 s. Find the net work done on the box.
(II) Eight books, each 4.3 cm thick with mass 1.7 kg, lie flat on a table. How much work is required to stack them one on top of another?
(II) (a) Find the force required to give a helicopter of mass M an acceleration of 0.10 g upward.
(b) Find the work done by this force as the helicopter moves a distance h upward.

PROBLEM SOLVING Kinetic Energy

(I) At room temperature, an oxygen molecule, with mass of 5.3 x 10^-26kg typically has a ke of abou 3.12 x 10^-21. How fast is the molecule moving?
(I) (a) If the ke of an arrow is doubled, by what factor has its speed increased? (b) If its speed is doubled, by what factor does its ke increase?
(I) How much work is required to stop a 3 kg object, which is moving with a speed of 19 m/s?
(I) How much work must be done to stop a 1250-kg car traveling at 105 km/hr?
(II) An 88-g arrow is fired from a bow whose string exerts an average force of 110 N on the arrow over a distance of 78 cm. What is the speed of the arrow as it leaves the bow?
(II) A baseball (m=140g) traveling 32 m/s moves a fielder’s glove backward 25 cm when the ball is caught. What was the average force exerted by the ball on the glove?
(II) If the speed of a car is increased by 50%, by what factor will its minimum braking distance be increased, assuming all else is the same? Ignore the driver’s reaction time.
(II) At an accident scene on a level road, investigators measure a car’s skid mark to be 88 m long. The accident occurred on a rainy day, and the coefficient of kinetic friction was estimated to be 0.42. Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. (Why does the car’s mass not matter?)
(II) A softball having a mass of 0.25 kg is pitched at 9 km/hr, By the time it reaches the plate, it may have slowed by 10%. Neglecting gravity, estimate the average force of air resistance during a pitch, if the distance between the plate and the pitcher is about 15 m.

PROBLEM SOLVING Potential Energy

(II) How high will a 1.85-kg rock go if thrown straight up by someone who does 80.0 J of work on it? Neglect air resistance.
(III) A 285-kg load is lifted 22.0 m vertically with an acceleration a=0.160g by a single cable. Determine (a) the tension in the cable, (b) the net work done on the load, (c) the work done by the cable on the load, (d) the work done by gravity on the load, and (e) the final speed of the load assuming it started from rest.
(I) A spring has a spring stiffness constant, k, of 440 N/m. How much must this spring be stretched to store 25 J of potential energy?
(I) A 7.0-kg monkey swings from one branch to another 1.2 m higher. What is the change in potential energy?
(I) By how much does the gravitational potential energy of a 64-kg pole vaulter change if his center of mass rises about 4.0 m during the jump?
(II) A 1200-kg car rolling on a horizontal surface has speed v= 65 km/hr when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring stiffness constant of the spring?
(II) A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to (a) the ground, and (b) the top of the person’s head? (c) How is the work done by the person related to the answers in parts (a) and (b)?
(II) A 55-kg hiker starts at an elevation of 1600 m and climbs to the top of a 3300-m peak. (a) What is the hiker’s change in potential energy? (b) What is the minimum work required of the hiker? (c) Can the actual work done be more than this? Explain why.
(II) A spring with k = 53 N/m hangs vertically next to a ruler. The end of the spring is next to the 15-cm mark on the ruler. If a 2.5-kg mass is now attached to the end of the spring, where will the end of the spring line up with the ruler marks?

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Sample Problem Power

PROBLEM SOLVING Power

(I) How long will it take a 1750-W motor to lift a 315-kg piano to a sixth-story window 16.0 m above?
(I) If a car generates 18 hp when traveling at a steady 88 km/hr. what must be the average force exerted on the car due to friction and air resistance?
(I) A 1400-kg sports car accelerates from rest to 95 km/hr in 7.4 s. What is the average power delivered by the engine?
(II) Electric energy units are often expressed in the form of “kilowatt-hours.” (a) Show that one kilowatt-hour (kWh) is equal to 3,600,000 J (b) If a typical family of four uses electric energy at an average rate of 520 W, how many kWh would their electric bill be for one month, and (c) how many joules would this be? (d) At a cost of P 7.00 per kWh, what would their monthly bill be in pesos? Does the monthly bill depend on the rate at which they use the electric energy?
(II) A driver notices that her 1150-kg car slows down from 85 km/hr to 65 km/hr in about 6.0 s on the level when it is in neutral. Approximately what power (watts and hp) is needed to keep the car traveling at a constant 75 km/hr?
(II) How much work can a 3.0-hp motor do in 1.0 h?
(II) A shot-putter accelerates a 7.3-kg shot from rest to 14 m/s If this motion takes 1.5 s, what average power was developed?
(II) A pump is to lift 18.0 kg of water per minute through a height of 3.60 m. What output rating (watts) should the pump motor have?
(II) During a workout, the football players at State U. ran up the stadium stairs in 66 s. The stairs are 140 m long and inclined at an angle of 32º. If a typical player has a mass of 95 kg, estimate the average power output on the way up. Ignore friction and air resistance.

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Sample Problem Momentum

PROBLEM SOLVING Momentum

(I) What is the magnitude of the momentum of a 28-g sparrow flying with a speed of 8.4 m/s?
(I) A constant friction force of 25 N acts on a 65-kg skier for 20 s. What is the skier’s change in velocity?
(II) A 0.145-kg baseball pitched at 39.0 m/s is hit on a horizontal line drive straight back toward the pitcher at 52.0 m/s If the contact time between bat and ball is 0.003 s, calculate the average force between the ball and bat during contact.
(II) A child in a boat throws a 6.40-kg package out horizontally with a speed of 10.0 m/sCalculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 26.0 kg, and that of the boat is 45.0 kg. Ignore water resistance.
(II) Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of 1500 kg/s with a speed of 40,000 m/s (at the moment of takeoff).
(II) A 95-kg halfback moving at 4.1 m/s on an apparent breakaway for a touchdown is tackled from behind. When he was tackled by an 85-kg cornerback running at 5.5 m/s in the same direction, what was their mutual speed immediately after the tackle?
(II) A 12,600-kg railroad car travels alone on a level frictionless track with a constant speed of 18.0 m/s A 5350-kg load, initially at rest, is dropped onto the car. What will be the car’s new speed?
(II) A 9300-kg boxcar traveling at 15.0 m/s strikes a second boxcar at rest. The two stick together and move off with a speed of 6.0 m/s. What is the mass of the second car?
(II) A 3800-kg open railroad car coasts along with a constant speed of 8.6 m/s on a level track. Snow begins to fall vertically and fills the car at a rate of 3.5 kg/min Ignoring friction with the tracks, what is the speed of the car after 90.0 min?
(II) A 23-g bullet traveling 230.0 m/s penetrates a 2.0-kg block of wood and emerges cleanly at 170.0 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?
(III) A 950-kg two-stage rocket is traveling at a speed of 5000 m/s with respect to Earth when a pre-designed explosion separates the rocket into two sections of equal mass that then move at a speed of 2200 m/s relative to each other along the original line of motion. (a) What are the speed and direction of each section (relative to Earth) after the explosion? (b) How much energy was supplied by the explosion? [Hint: What is the change in ke as a result of the explosion?]
(II) A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s The golf club was in contact with the ball for 3.5 x 10^-3s. Find (a) the impulse imparted to the golf ball, and (b) the average force exerted on the ball by the golf club.
(II) A 12-kg hammer strikes a nail at a velocity of 8.5 m/s and comes to rest in a time interval of 8.0 ms. (a) What is the impulse given to the nail? (b) What is the average force acting on the nail?
(II) A tennis ball of mass m=0.060 kg and speed v = 25 m/s strikes a wall at a 45º angle and rebounds with the same speed at 45º. What is the impulse (magnitude and direction) given to the ball?
(II) You are the design engineer in charge of the crashworthiness of new automobile models. Cars are tested by smashing them into fixed, massive barriers at 50 km/hr (30 mph). A new model of mass 1500 kg takes 0.15 s from the time of impact until it is brought to rest. (a) Calculate the average force exerted on the car by the barrier. (b) Calculate the average deceleration of the car.
(II) A 95-kg fullback is running at 4.0 m/s to the east and is stopped in 0.75 s by a head-on tackle by a tackler running due west. Calculate (a) the original momentum of the fullback, (b) the impulse exerted on the fullback, (c) the impulse exerted on the tackler, and (d) the average force exerted on the tackler.

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