The Big Idea
The universe has many remarkable qualities, among them a rather beautiful symmetry: the total amount of motion in the universe balances out … always. This law only makes sense if we measure “motion” in a specific way: as the product of mass and velocity. This product, called momentum, can be exchanged from one object to another in a collision. The rapidity with which momentum is exchanged over time is determined by the forces involved in the collision. Conservation of momentum is one of the five fundamental conservation laws in the universe. The other four are conservation of energy, angular momentum, charge conservation and CPT conservation.
p=mv - Momentum equals mass multiplied by velocity. It is a vector, direction must be considered. Always define a positive direction and be consistent.
Δp=pf−pi - Impulse is defined as the difference between the final momentum and
the initial momentum of an object. When calculating impulse, consider the change in momentum of one of the objects in the collision.
F=△p△t - Note that ΔpΔt=mΔvΔt=ma. This is just Newton’s 2nd Law!
∑pi=∑pf - The sum of all the momenta before a collision equals the sum of all the momenta after the collision. When adding up momenta, be sure to add them as vectors, so include the positives and negatives and if a two dimensional collision, break into the x and y components. Also be sure to include all the objects in the system.
- Momentum is a vector that points in the direction of the velocity vector. The magnitude of this vector is the product of mass and speed.
- The total momentum of the universe is always the same and is equal to zero. The total momentum of an isolated system never changes.
- Momentum can be transferred from one body to another. In an isolated system in which momentum is transferred internally, the total initial momentum is the same as the total final momentum.
- Momentum conservation is especially important in collisions, where the total momentum just before the collision is the same as the total momentum after the collision.
- The force imparted on an object is equal to the change in momentum divided by the time interval over which the objects are in contact.
- Two cars collide head-on … two subatomic particles collide in an accelerator … a bird slams into a glass office building: all of these are examples of one-dimensional (straight line) collisions. For these, pay extra attention to direction: define one direction as positive and the other as negative, and be sure everybody gets the right sign.
- A firecracker in mid-air explodes … two children push off each other on roller skates … an atomic nucleus breaks apart during a radioactive decay: all of these are examples of disintegration problems. The initial momentum beforehand is zero, so the final momentum afterwards must also be zero.
- A spacecraft burns off momentum by colliding with air molecules as it descends … hail stones pummel the top of your car … a wet rag is thrown at and sticks to the wall: all of these are examples of impulse problems, where the change in momentum of one object and the reaction to the applied force are considered. What is important here is the rate: you need to come up with an average time △t that the collision(s) last so that you can figure out the force F=△p△t. Remember as well that if a particle has momentum p, and it experiences an impulse that turns it around completely, with new momentum −p, then the total change in momentum has magnitude 2p. It is harder to turn something totally around than just to stop it!
- A car going south collides with a second car going east … an inflatable ball is thrown into the flow of a waterfall … a billiard ball strikes two others, sending all three off in new directions: these are all examples of two-dimensional (planar) collisions. For these, you get a break: the sum of all the momenta in the x direction have to remain unchanged before and after the collision — independent of any y momenta, and vice-versa. This is a similar concept to the one we used in projectile motion. Motions in different directions are independent of each other.
Momentum Conservation Problem Set
- You find yourself in the middle of a frozen lake. There is no friction between your feet and the ice of the lake. You need to get home for dinner. Which strategy will work best?
- Press down harder with your shoes as you walk to shore.
- Take off your jacket. Then, throw it in the direction opposite to the shore.
- Wiggle your butt until you start to move in the direction of the shore.
- Call for help from the great Greek god Poseidon.
- You jump off of the top of your house and hope to land on a wooden deck below. Consider the following possible outcomes: I. You hit the deck, but it isn’t wood! A camouflaged trampoline slows you down over a time period of 0.2 seconds and sends you flying back up into the air. II. You hit the deck with your knees locked in a straight-legged position. The collision time is 0.01 seconds. III. You hit the deck and bend your legs, lengthening the collision time to 0.2 seconds. IV. You hit the deck, but it isn’t wood! It is simply a piece of paper painted to look like a deck. Below is an infinite void and you continue to fall, forever.
- Which method will involve the greatest force acting on you?
- Which method will involve the least force acting on you?
- Which method will land you on the deck in the least pain?
- Which method involves the least impulse delivered to you?
- Which method involves the greatest impulse delivered to you?
- You and your sister are riding skateboards side by side at the same speed. You are holding one end of a rope and she is holding the other. Assume there is no friction between the wheels and the ground. If your sister lets go of the rope, how does your speed change?
- It stays the same.
- It doubles.
- It reduces by half.
- You and your sister are riding skateboards (see Problem 3), but now she is riding behind you. You are holding one end of a meter stick and she is holding the other. At an agreed time, you push back on the stick hard enough to get her to stop. What happens to your speed? Choose one. (For the purposes of this problem pretend you and your sister weigh the same amount.)
- It stays the same.
- It doubles.
- It reduces by half.
- You punch the wall with your fist. Clearly your fist has momentum before it hits the wall. It is equally clear that after hitting the wall, your fist has no momentum. But momentum is always conserved! Explain.
- An astronaut is using a drill to fix the gyroscopes on the Hubble telescope. Suddenly, she loses her footing and floats away from the telescope. What should she do to save herself?
- You look up one morning and see that a 30 kg chunk of asbestos from your ceiling is falling on you! Would you be better off if the chunk hit you and stuck to your forehead, or if it hit you and bounced upward? Explain your answer.
- Standing on a skateboard, Joe throws a bowling ball forward and as a result he rolls backwards. If instead Joe holds onto the ball instead of releasing it as he “throws”, will he still recoil? Explain.
- Waiting for her mom to come out of the store, an impatient little girl is sitting in a car pushing on the dashboard. Is this dangerous – could the little girl set the car into motion by pushing on the dashboard? Explain – discuss internal vs. external forces.
- Explain how a rocket can travel through space with no land or even air to push against.
- There’s a boxing term called “roll with the punches” - if you’re in the ring with a boxer with a fast punch, it could save your life. What’s meant by that phrase, and why does such a maneuver decrease the force of the punch?
- Do some online research and find out what a car’s “crumple zone” is, and how it helps reduce forces in a crash Use the words change of momentum, impulse, and force in your answer.
- Explain why it takes several firefighters to hold on to a fire house spraying a large volume of water at a high flow rate.
- Explain why a gun recoils when it’s fired, and why a larger caliber bullet will produce more recoil velocity. Will a starter’s pistol (firing blanks) also recoil? Explain.
- During the Gold Rush, an inventor named Pelton got rich by inventing an improved water wheel. Go online and find out what his innovation was, and explain how making the water “bounce” exerted more force on the wheel, and thus a more efficient device.
- A 5.00 kg firecracker explodes into two parts: one part has a mass of 3.00 kg and moves at a velocity of 25.0 m/s towards the west. The other part has a mass of 2.00 kg. What is the velocity of the second piece as a result of the explosion?
- A firecracker lying on the ground explodes, breaking into two pieces. One piece has twice the mass of the other. What is the ratio of their speeds?
- David was driving too fast in his PT Cruiser and crashed into a tree. He was originally going 35 km/hr. Assume it took 0.4 seconds for the tree to bring him to a stop. The mass of the driver and the car is 450 kg.
- What average force did he experience? Include a direction in your answer.
- What average force did the tree experience? Include a direction in your answer.
- An astronaut is 100 m away from her spaceship doing repairs with a 10.0 kg wrench. The astronaut’s total mass is 90.0 kg and the ship has a mass of 1.00×104 kg. If she throws the wrench in the opposite direction of the spaceship at 10.0 m/s.
- calculate her recoil velocity
- How long would it take for her to reach the ship?
- A baseball player faces an 80.0 m/s pitch. In a matter of .020 seconds he swings the bat, hitting a 50.0 m/s line drive back at the pitcher. Calculate the force on the bat while in contact with the ball. (HINT – remember velocity is a vector!) The mass of the ball is 0.5 kg.
- You throw your 6.0 kg skateboard down the street, giving it a speed of 4.0 m/s. Your friend, the Frog, jumps on your skateboard from rest as it passes by. Frog has a mass of 60 kg.
- What is the momentum of the skateboard before Frog jumps on it?
- Find Frog’s speed after he jumps on the skateboard.
- What impulse did Frog deliver to the skateboard?
- If the impulse was delivered over 0.2 seconds, what was the average force imparted to the skateboard?
- What was the average force imparted to Frog? Explain.
- While driving in your pickup truck down Highway 280 between San Francisco and Palo Alto, an asteroid lands in your truck bed! Despite its 220 kg mass, the asteroid does not destroy your 1200 kg truck. In fact, it landed perfectly vertically.
- Before the asteroid hit, you were going 25 m/s. After it hit, how fast were you going?
- You’re so unnerved by the asteroid collision that you neglect to see the “Bridge Out” sign at Crystal Springs and you plunge horizontally off the freeway into the gulley below. If the bridge is 85 m high and you’re traveling at the speed found in (a), how far horizontally will you travel before hitting the ground below? Draw a sketch and label distances and velocities to assist in your solution.
- Two blocks collide on a frictionless surface, as shown. Afterwards, they have a combined mass of 10 kg and a speed of 2.5 m/s. Before the collision, one of the blocks was at rest. This block had a mass of 8.0 kg.
- What was the mass and initial speed of the second block?
- After the collision the blocks stick and slide; they’re traveling at 2.5 m/s when they hit a rough patch on the surface. If the force of friction on the blocks is 6.3 N,
- Draw a FBD for the blocks (treat the blocks as one mass)
- Calculate the acceleration of the blocks
- Calculate how far the blocks will slide before coming to rest.
- Larry and Jerry are riding together in an amusement park bumper car at 10 m/s, with a total mass = 300 kg. They bump Marissa’s car, mass 125 kg, which is sitting still. Assume an elastic collision.
(a) Fill in the table below with each momentum and the total momentum before the collision.
Momentum before (kg m/s)
Momentum after (kg m/s)
Larry and Jerry car
(b) After the collision, the boys continue ahead with a speed of 4 m/s. Calculate their new momentum and enter it into the table. Then use conservation of momentum to complete the table and calculate Marissa’s velocity after the collision.
- Running at 2.0 m/s, Thrack, the 45.0-kg quarterback, collides with Biff, the 90.0-kg tackle, who is traveling at 6.0 m/s in the other direction. Assume Thrack is running to the right and Biff is running to the left.
Momentum before (kg m/s)
Momentum after (kg m/s)
TOTAL MOMENTUM OF SYSTEM
(a) Fill in the “before” side of the table.
(b) Upon collision, Biff continues to travel forward (i.e. to the left) at 1.0 m/s. Calculate his new momentum, enter it in the table, and then use conservation of momentum to calculate Thrack’s new momentum. In which direction is he moving?
(c) Calculate Thrack’s new velocity.
- Serena Williams volleys a tennis ball hit to her at 30 m/s. She blasts it back to the other court at 50 m/s. A standard tennis ball has mass of 0.057 kg. If Serena applied an average force of 500 N to the ball while it was in contact with the racket, how long did the contact last?
- Joe, mass 85 kg, is eating peanut butter and riding his skateboard north down the street at 12 m/s when he collides head-on with Janet, mass 65 kg, who is eating chocolate and roller skating at 9 m/s south. Neither of them are paying attention – they collide and stick. Calculate the magnitude and direction of their resultant velocity. A diagram may be helpful.
- In the above picture, the carts are moving on a level, frictionless track. After the collision all three carts stick together. Determine the direction and speed of the combined carts after the collision. (Assume 3-significant digit accuracy.)
- Zoran’s spacecraft, with mass 12,000 kg, is traveling to space. The structure and capsule of the craft have a mass of 2,000 kg; the rest is fuel. The rocket shoots out 0.10 kg/s of fuel particles with a velocity of 700 m/s with respect to the craft.
- What is the acceleration of the rocket in the first second?
- What is the change in velocity and the average acceleration of the rocket during the first ten minutes have passed?
- 37.5 m/s
- 11250 N to the left
- tree experienced same average force of 11000 N but to the right
- 1.1 m/s
- 90 seconds
- 3250 N
- 0.364 m/s
- 109 N
- 109 N due to Newton’s third law
- 21 m/s
- 87 m
- 2.0 kg, 12.5 m/s
0.63 m/s2,5 m
- 3000 kg m/s, 0,3000 kg m/s
- 14.4 m/s
- 90 kg m/s (right), -540 kg m/s (left), -450 kg m/s
- -90 kg m/s, -360 kg m/s and moving to the left
- -8 m/s
- 0.0091 s
- 2.9 m/s, north
- 0.13 m/s to the left
3.5 m/s,0.0059 m/s2
- In Sacramento a 4000 kg SUV is traveling 30 m/s south on Truxel crashes into an empty school bus, 7000 kg traveling east on San Juan at 15 m/s. The SUV and truck stick together after the collision.
- Find the velocity of the wreck just after collision
- Find the direction in which the wreck initially moves
- A 3 kg ball is moving 2 m/s in the positive x−direction when it is struck dead center by a 2 kg ball moving in the positive y−direction at 1 m/s. After collision the 3 kg ball moves at 3 m/s 30 degrees from the positive x−axis.
(a) To 2-significant digit accuracy fill out the following table:
3 kg ball px
3 kg ball py
2 kg ball px
2 kg ball py
Momentum after collision
(b) Find the velocity and direction of the 2 kg ball.
(c) Use the table to prove momentum is conserved.
- The train engine and its four boxcars are coasting at 40 m/s. The engine train has mass of 5,500 kg and the boxcars have masses, from left to right, of 1,000 kg, 1,500 kg, 2,000 kg, and 3,000 kg. (For this problem, you may neglect the small external forces of friction and air resistance.)
- What happens to the speed of the train when it releases the last boxcar? (Hint: think before you blindly calculate.)
- If the train can shoot boxcars backwards at 30 m/s relative to the train’s speed, how many boxcars does the train need to shoot out in order to obtain a speed of 58.75 m/s?
Answers to Optional Problems:
- (b) 1.8 m/s 16∘
- (b) 4.6 m/s 68∘
- no change
- the last two cars