Physics- Momentum In An Explosion Homework Sheet With Answers PDF

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HW7: Momentum Due: 11:59pm on Wednesday, April 11, 2018 *EXT* You will receive no credit for items you complete after the assignment is due. Grading Policy

Momentum in an Explosion A giant "egg" explodes as part of a fireworks display. The egg is at rest before the explosion, and after the explosion, it breaks into two pieces, with piece B moving in the positive x direction. The masses of both pieces are indicated in , shown traveling in opposite directions.

Part A What is the magnitude of the momentum |pA,i | of piece A before the explosion? Express your answer numerically in kilogram meters per second.

Hint 1. Initial momentum The momentum of any object is determined by the product of the object's mass and velocity. The egg is initially at rest. Use this to find the initial momentum.

ANSWER: |p A,i | = 0

kg ⋅ m/s

Correct Similarly, piece B has zero momentum before the collision. The total momentum of the "egg," the sum of the two individual momenta, is also zero.

Part B During the explosion, is the magnitude of the force of piece A on piece B greater than, less than, or equal to the magnitude of the force of piece B on piece A?

Hint 1. Forces in an explosion The forces specified in this problem must obey Newton's third law, which states that every action has an equalmagnitude and oppositely-directed reaction.

ANSWER:

greater than less than equal to cannot be determined

Correct

Part C The component of the momentum of piece B, p Bx,f , is measured to be +500 kg ⋅ m/s after the explosion. Find the component of the momentum p Ax,f of piece A after the explosion. Enter your answer numerically in kilogram meters per second.

Hint 1. Conservation of momentum The law of conservation of momentum states that the total momentum in an isolated system of objects must remain constant, regardless of the interactions (or collisions) between the objects. Thus, the total momentum of the two pieces of the egg after the explosion must be equal to the total momentum of the two pieces of the egg before the explosion.

ANSWER:

p Ax,f = -500 kg ⋅ m/s Correct

Problem 8.05 The speed of the fastest-pitched baseball was 42.0 m/s , and the ball's mass was 145 g .

Part A What was the magnitude of the momentum of this ball?

ANSWER:

p = 6.09 kg ⋅ m/s Correct

Part B How many joules of kinetic energy did this ball have? ANSWER:

K = 128 J Correct

Part C How fast would a 50.0 gram ball have to travel to have the same amount of kinetic energy? ANSWER:

v = 71.5 m/s Correct

Part D How fast would a 50.0 gram ball have to travel to have the same amount of momentum? ANSWER:

v = 122 m/s Correct

Problem 8.09: Recoil Speed of the Earth In principle, any time someone jumps up, the earth moves in the opposite direction.

Part A To see why we are unaware of this motion, calculate the recoil speed of the earth when a 81.0 kg person jumps upward at a speed of 2.10 m/s . Consult Appendix E in the textbook as needed.

Express your answer using two significant figures. ANSWER:

v = 2.8×10−23 m/s Correct

Collisions in One Dimension On a frictionless horizontal air table, puck A (with mass 0.248 kg ) is moving toward puck B (with mass 0.371 kg ), which is initially at rest. After the collision, puck A has velocity 0.118 m/s to the left, and puck B has velocity 0.655 m/s to the right.

Part A What was the speed vAi of puck A before the collision?

Hint 1. How to approach the problem Apply the conservation of momentum equation. Keep in mind that momentum is a vector quantity, with both magnitude and direction. In a one-dimesional problem like this one, the direction of the momentum vector can be indicated by its sign. For the subparts that follow, take the positive direction to be to the right, the initial direction of puck A.

Hint 2. The initial momentum The initial momentum p Ai is given by

p Ai = ( 0.248 kg )vAi . Hint 3. Find the final momentum of puck A Find the final momentum p Af of puck A, including the sign. Assume that the positive x direction is to the right. ANSWER:

p Af = −2.93×10−2 kg ⋅ m/s

Hint 4. Find the final momentum of puck B Find the final momentum p Bf of puck B, including the sign. Assume that the positive x direction is to the right. ANSWER:

p Bf = 0.243 kg ⋅ m/s

ANSWER:

vAi = 0.862 m/s Correct If you are required to use the answer obtained for a subsequent hint or part, use your unrounded/full precision answer.

Part B Calculate ∆K , the change in the total kinetic energy of the system that occurs during the collision.

Hint 1. How to approach the problem Use the velocity and mass of each puck before and after the collision to find their respective kinetic energies. The change in kinetic energy is then the final kinetic energy of the system minus the initial kinetic energy.

Hint 2. Find the initial kinetic energy of puck A Find K Ai , the initial kinetic energy of puck A. Express your answer in joules. ANSWER:

K Ai = 9.21×10−2 J

Hint 3. Find the final kinetic energy of puck A Find K Af , the final kinetic energy of puck A. ANSWER:

K Af = 1.73×10−3 J

Hint 4. Find the final kinetic energy of puck B Find K Bf , the final kinetic energy of puck B. ANSWER:

K Bf = 7.96×10−2 J

ANSWER:

∆K = −1.08×10−2 J

All attempts used; correct answer displayed

± Catching a Ball on Ice Olaf is standing on a sheet of ice that covers the football stadium parking lot in Buffalo, New York; there is negligible friction between his feet and the ice. A friend throws Olaf a ball of mass 0.400 kg that is traveling horizontally at 10.8m/s . Olaf's mass is 72.9 kg .

Part A If Olaf catches the ball, with what speed v f do Olaf and the ball move afterward? Express your answer numerically in meters per second.

Hint 1. How to approach the problem Using conservation of momentum and the fact that Olaf's initial momentum is zero, set the initial momentum of the ball equal to the final momentum of Olaf and the ball, then solve for the final velocity.

Hint 2. Find the ball's initial momentum What is p i , the initial momentum of the ball? Express your answer numerically in kilogram meters per second. ANSWER:

p i = 4.32

ANSWER:

kg⋅m s

v f = 5.89×10−2 m/s All attempts used; correct answer displayed

Part B If the ball hits Olaf and bounces off his chest horizontally at 8.90m/s in the opposite direction, what is his speedv f after the collision? Express your answer numerically in meters per second.

Hint 1. How to approach the problem The initial momentum of the ball is the same as in Part A. Apply conservation of momentum, keeping in mind that both Olaf and the ball have a nonzero final momentum.

Hint 2. Find the ball's final momentum

 , the ball's final Taking the direction in which the ball was initially traveling to be positive, what is p ball,f momentum? Express your answer numerically in kilogram meters per second. ANSWER:

 = -3.56 p ball,f

kg⋅m s

ANSWER:

v f = 0.108 m/s Correct

PSS 8.1: Bird Defense Learning Goal: To practice Problem-Solving Strategy 8.1 Conservation of momentum. To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a 600 g falcon flying at 20.0 m/s ran into a 1.50 kg raven flying at 9.00 m/s . The falcon hit the raven at a right angle to its original path and bounced back with a speed of 5.00 m/s . By what angle did the falcon change the raven's direction of motion? Problem-Solving Strategy 8.2 Conservation of momentum SET UP

1. Define the system you are analyzing and choose a coordinate system. Often, it is easiest to choose the x axis as having the direction of one of the initial velocities. Make sure that you are using an inertial frame of reference. 2. Treat each object as a particle. Sketch "before" and "after" diagrams. Add vectors on each diagram to represent all known velocities and label their magnitudes and angles. Give each unknown quantity an algebraic symbol. For example, the initial value (i) of the x component of velocity of object A can be written as v A,i,x . SOLVE

3. Compute the x and y components of momentum of each particle, both before and after the interaction, using the relations p x = mv x and p y = mvy . Some of the components will be expressed in terms of symbols representing unknown quantities. Note that the components of velocity and momentum can be either positive or negative, so be careful with signs. 4. Determine whether both the x and the y components of total momentum are conserved. If they are, write a relation that equates the total initial x component of momentum to the total final x component of momentum, and similarly for the y components. These two equations express conservation of momentum in component form. In some problems, only one component of momentum is conserved. 5. Solve the equations you wrote in step 4 to determine whatever quantities the problem asks for. In some problems, you will have to convert from the x and y components of a velocity to its magnitude and direction, and in others you will need to make the opposite conversion. REFLECT

6. Try to think of particular cases in which you can guess what the results of your analysis should be. What happens when two masses are equal? When one mass is zero? Is the result what you would expect?

SET UP Before writing any equations, organize your information and draw appropriate diagrams.

Part A Which set of axes shown in the figure represents the best orientation for the coordinate axes?

ANSWER:

Set A Set B Set C Set D

Correct Note that no information is given on the direction of motion of the two birds before the collision, except that they collide at a right angle. Any two directions mutually perpendicular are equally correct. Whatever configuration you choose, however, it is generally easiest to choose the axes as having the direction of the initial velocities.

Part B We can make a sketch of the system before the collision by simplifying the picture selected in Part A and replacing each object with a particle of equal mass. The resulting sketch is shown in the following figure, where particle F represents the falcon with initial velocity vFi and particle R represents the raven with initial velocity vRi . Now consider the following four diagrams, where vRf and v Ff are the final velocities of the raven and of the falcon. Which one

correctly shows the system after the collision? ANSWER:

Set A Set B Set C Set D

Correct

SOLVE Now that you have set up the problem, choose appropriate equations and solve for your unknowns.

Part C Write down the initial x and y components of the momentum of the falcon: p Fi,x andp Fi,y . Then, write the initial x and y components of the momentum of the raven: p Ri,x and p Ri,y . Note that the initial components of momentum are the components of momentum before the collision. Express your answers p Fi,x , p Fi,y , p Ri,x , and p Ri,y in this specific order and separated by commas, in kilogram meters per second.

Hint 1. Components of momentum

 a particle of mass m and velocity v is a vector quantity defined as p The momentum p of components are

 mv . Its x and y =

p x = mv x and p y = mv y, where v x and v y are the x and the y components of velocity.

Hint 2. Find the initial components of velocity of the falcon What are v Fi,x and v Fi,y , respectively, the initial x and y components of velocity of the falcon? Express your answers, separated by a comma, in meters per second. ANSWER:

v Fi,x , v Fi,y = 0,20.0 m/s

Hint 3. Find the initial components of velocity of the raven What are v Ri,x and v Ri,y , the initial x and y components of velocity of the raven? Express your answers, separated by a comma, in meters per second. ANSWER:

v Ri,x , v Ri,y = 9.00,0 m/s

ANSWER: 0,12.0,13.5,0

kg ⋅ m/s

Correct

Part D Give the final x and y components of the momentum of the falcon:p Ff,x andp Ff,y . Note that the final components of momentum are the components of momentum after the collision. Express your answers p Ff,x and p Ff,y separated by commas in kilogram meters per second.

Hint 1. Components of momentum

 a particle of mass m and velocity v is a vector quantity defined as p The momentum p of components are

 mv . Its x and y =

p x = mv x and p y = mv y, where v x and v y are the x and the y components of velocity.

Hint 2. Find the final components of velocity of the falcon What are v Ff,x and v Ff,y , the final x and y components of velocity of the falcon? Express your answers, separated by a comma, in meters per second. ANSWER:

v Ff,x , v Ff,y = 0,-5.00 m/s

ANSWER: 0,-3.00

kg ⋅ m/s

Correct

Part E Give the final x and y components of the momentum of the raven: p Rf,x andp Rf,y . Note that the final components of momentum are the components of momentum after the collision. Since these quantities are unknown, you will need to use symbols to express the unknown terms. In particular, use the symbol θ for the angle between the raven's direction of motion after the collision and the positive x axis, andv Rf for the raven's final speed. Express your answers p Rf,x andp Rf,y separated by commas using the variablesθ , for the angle that the raven's direction of motion makes with the positive x axis, andv Rf , for the raven's final speed.

Hint 1. Components of momentum



 a particle of mass m and velocity v is a vector quantity defined as p The momentum p of components are

 mv . Its x and y =

p x = mv x and p y = mv y, where v x and v y are the x and the y components of velocity.

Hint 2. Find the final components of velocity of the raven Write an expression for v Rf,x and v Rf,y , the final x and y components of velocity of the raven. Since some or all of these quantities are unknown, you will need to use symbols to express the unknown terms. In particular, use θ for the angle that the raven's direction of motion makes with the positive x axis, andv Rf for the raven's final speed. Express your answers, separated by a comma, in terms of some or all the variablesθ andv Rf . ANSWER:

v Rf,x , v Rf,y = v Rf cos(θ),v Rf sin(θ)

ANSWER:

1.5v Rf cos(θ),1.5vRf sin(θ) Answer Requested

Part F Which of the following statements correctly applies to the system that you are analyzing? ANSWER:

Only the x component of total momentum is conserved, and one can write the following expression:

p Fi,x + p Ri,x = pFf,x + p Rf,x . Only the y component of total momentum is conserved, and one can write the following expression:

p Fi,y + p Ri,y = pFf,y + p Rf,y . Both the x and the y components of total momentum are conserved, and one can write the following expression: p Fi,x + p Ri,x + p Fi,y + p Ri,y = p Ff,x + p Rf,x + p Ff,y + p Rf,y . Both the x and the y components of total momentum are conserved, and one can write the following expressions: p Fi,x + p Ri,x = p Ff,x + p Rf,x and p Fi,y + p Ri,y = p Ff,y + p Rf,y . None of the components of total momentum are conserved because gravity acts on the system and the system is not isolated.

Correct Now substitute the expressions for each component of momentum found in the previous parts. If you do so, you will obtain a system of two linear equations with θ and v Rf as unknowns.

Part G By what angle θ did the falcon change the raven's direction of motion? Express your answer in degrees to two significant figures.

Hint 1. How to approach the problem In the previous part you found that both the x and the y components of total momentum are conserved. Thus, you can write two equations,

p Fi,x + p Ri,x = p Ff,x + p Rf,x , p Fi,y + p Ri,y = p Ff,y + p Rf,y , that express conservation of momentum in component form. These two expressions are also a system of two linear equations in θ and v Rf . Take the first equation and solve for v Rf . Substitute your result in the second equation and solve for θ .

Hint 2. Find an expression for the final speed of the raven Which of the following expressions can be derived from the conservation of the x component of total momentum? Let v Ri and mR be the initial speed and the mass of the raven. ANSWER: v Ri cos θ cos θ v Ri

v Rf =

m Rv Ri cos θ cos θ m Rv Ri v Ri m R cos θ m R cos θ v Ri

ANSWER:

θ = 48 degrees

Correct

REFLECT Think about whether your results make sense.

Part H Despite being only about one-third the mass of the raven, the falcon successfully diverts the raven's direction of motion by a relatively large angle. This is because of the initial high speed of the falcon. In fact, by what angleθ would the falcon change the raven's direction of motion if it were to hit the raven at a lower speed, for example at a speed equal to the raven's initial speed? Assume that the raven has the same final speed as above and the falcon has the same final velocity as given in the introduction. Express your answer in degrees to two sigificant figures.

Hint 1. Find the initial momentum of the falcon What is the initial momentum of the falcon, p Fi if it flies at a speed 9.00 m/s ? Express your answer in kilogram meters per second. ANSWER:

p Fi = 5.40 kg ⋅ m/s

ANSWER:

θ = 32 degrees All attempts used; correct answer displayed If the falcon were to hit the raven at a lower speed, the raven's direction of motion would change by a smaller angle, allowing the raven to continue its motion towards the falcon's nest with little disturbance. In such circumstance, the defense technique of the falcon would not be very effective.

A Girl on a Trampoline A girl of mass m1 second. At height

= 60.0 kilograms springs from a trampoline with an initial upward velocity ofv i = 8.00 meters per h = 2.00 meters above the trampoline, the girl grabs a box of mass m2 = 15.0 kilograms.

For this problem, use g

= 9.80 meters per second per second for the magnitude of the acceleration due to gravity.

Part A What is the speed v before of the girl immediately before she grabs the box? Express your answer numerically in meters per second.

Hint 1. How to approach the problem Use conservation of energy. Find the initial kinetic energy Ki of the girl as she leaves the trampoline. Then find her gravitational potential energy Ubefore just before she grabs the box (define her initial potential energy to be zero). According to the principle of conservation of energy, K i = Ubefore + K before . Once you have Kbefore , use the definition of translational kinetic energy to find the girl's speed v before .

Hint 2. Initial kinetic energy What is the girl's initial kinetic energy Ki as she leaves the trampoline? Express your answer numerically in joules. ANSWER:

K i = 1920 J

Hint 3. Potential energy at height h What is the girl's gravitational potential energy Ubefore immediately before she grabs the box? Express your answer numerically in joules. ANSWER:

Ubefore = 1180 J

ANSWER:

v before = 4.98 m/...