Linear momentum and impulse

A tennis ball of $0.06\ \text{kg}$ leaves the serve at $50\ \text{m/s}$.

1. a) Write the definition of linear momentum, $\vec{p} = m \cdot \vec{v}$, and state its SI unit.
2. b) Calculate the magnitude of the linear momentum of the ball.

A truck of $3000\ \text{kg}$ travels at $10\ \text{m/s}$ and a motorcycle with its rider, weighing $200\ \text{kg}$ in total, travels at $20\ \text{m/s}$. The diagram compares both vehicles.

1. a) Calculate the linear momentum of each vehicle.
2. b) How many times greater is the linear momentum of the truck than that of the motorcycle?
3. c) Explain why the truck, although traveling more slowly, is much harder to stop.

A runner with a mass of $60\ \text{kg}$ moves in a straight line toward the east at $3\ \text{m/s}$.

1. a) Calculate the magnitude of her linear momentum.
2. b) State the direction of the vector $\vec{p}$.
3. c) What must happen for her linear momentum to change? Relate this to the net force acting on the runner.

A cyclist and their bicycle have a total mass of $80\ \text{kg}$ and travel at $54\ \text{km/h}$.

1. a) Express the velocity in $\text{m/s}$.
2. b) Calculate the linear momentum of the system.
3. c) What impulse must be applied, as a minimum, to stop them? State its magnitude and direction.

When pushing a supermarket cart, an average force of $20\ \text{N}$ is exerted for $3\ \text{s}$.

1. a) Write the definition of mechanical impulse, $\vec{I} = \vec{F} \cdot \Delta t$, and state its unit.
2. b) Calculate the impulse received by the cart.

Impulse is measured in $\text{N} \times \text{s}$ and linear momentum in $\text{kg} \times \text{m/s}$.

1. a) Using the definition of a newton deduced from $\vec{F} = m \cdot \vec{a}$, show that $1\ \text{N} \times \text{s} = 1\ \text{kg} \times \text{m/s}$.
2. b) Explain why it makes physical sense for impulse and the change in linear momentum to be measured in the same unit.

A soccer ball of $0.4\ \text{kg}$, initially at rest on the ground, flies off at $15\ \text{m/s}$ during the kickoff of the game.

1. a) Calculate the linear momentum of the soccer ball before and after the kick.
2. b) Calculate the change in linear momentum of the soccer ball, $\Delta \vec{p}$.

In a volleyball match, a ball of $0.25\ \text{kg}$ reaches a player's hands at $10\ \text{m/s}$ and is returned in the opposite direction at $20\ \text{m/s}$. The contact with her hands lasts $0.05\ \text{s}$. The diagram shows the ball before and after contact.

1. a) Taking the direction in which the ball leaves as positive, calculate the change in its linear momentum (be careful with the signs of the velocities).
2. b) Apply the impulse-momentum theorem, $\vec{F} \cdot \Delta t = \Delta \vec{p}$, and calculate the average force that the player exerts on the ball.

A lab cart of $2\ \text{kg}$, at rest on a track, receives an impulse of $10\ \text{N} \times \text{s}$ from a sharp blow.

1. a) Applying $\vec{I} = \Delta \vec{p}$, calculate the velocity acquired by the cart.
2. b) What impulse would be needed to impart to it so that, starting from rest, it reached double the velocity? Explain the proportionality between impulse and final velocity.

A squash ball of $0.5\ \text{kg}$ hits a wall perpendicularly at $8\ \text{m/s}$ and bounces back at $6\ \text{m/s}$ in the opposite direction. The contact with the wall lasts $0.1\ \text{s}$.

1. a) Calculate the change in the linear momentum of the ball (be careful with the signs of the velocities).
2. b) Calculate the average force that the wall exerts on the ball.
3. c) By Newton's third law (action and reaction), what force does the ball exert on the wall during contact? State its magnitude and direction.

A driver of $70\ \text{kg}$ is traveling at $54\ \text{km/h}$ and suffers a head-on collision that stops the car almost instantly.

1. a) Express the velocity in $\text{m/s}$ and calculate the change in the driver's linear momentum.
2. b) The airbag takes $0.15\ \text{s}$ to stop the driver: calculate the average force it exerts on them.
3. c) Without an airbag, the driver's head, having a mass of about $5\ \text{kg}$, would hit the steering wheel and stop in barely $0.01\ \text{s}$: calculate the change in the linear momentum of the head and the average force it would withstand, and compare it with the force on the head if the airbag stopped it in $0.15\ \text{s}$.
4. d) In view of the results, explain the two functions of the airbag: what it achieves by increasing the braking time (relate it to $\vec{F} \cdot \Delta t = \Delta \vec{p}$) and how it distributes the force over the body.

In a crash test, a car of $1200\ \text{kg}$ traveling at $72\ \text{km/h}$ impacts a wall and comes to a stop in $0.1\ \text{s}$ (take $g = 10\ \text{m/s}^2$).

1. a) Express the velocity in $\text{m/s}$.
2. b) Calculate the change in the linear momentum of the car during the collision.
3. c) Calculate the average force that the wall exerts on the car.
4. d) Calculate the weight of the car and express the force of the collision as a "number of times the weight": explain why collisions at that speed are so destructive.

Unit challenge. How does engineering design impact-absorption systems? Become a road safety expert by analyzing the physics of a collision.

1. Home data collection: Determine the mass of a ball ($m \pm 0.1\text{ g}$). Drop it vertically from 5 different heights between $0.20\text{ m}$ and $1.00\text{ m}$ (tape measure uncertainty: $\pm 0.001\text{ m}$). Record with your cell phone to accurately measure the rebound height for each drop.

2. Data processing and graphing: Construct a data table showing the heights, velocities before and after the collision ($v = \sqrt{2gh}$), and the change in linear momentum $\Delta p = m(v_2 - (-v_1))$. Plot $\Delta p$ against the impact velocity $v_1$, including estimated error bars.

3. Impulse and technical proposal: Use slow-motion video ($120$ or $240\text{ fps}$) to measure the contact time $\Delta t$. Apply the impulse-momentum theorem ($I = F_{\text{mean}}\Delta t = \Delta p$) and Newton's 2nd law to deduce the average impact force. Repeat by dropping the ball onto a cushion, and write a brief comparative report justifying the use of crumple zones in vehicles.

Evaluation criteria:

• Rigor in experimental measurement, calculation of uncertainties, and preparation of the data table and graph.

• Scientific accuracy in applying the laws of dynamics, linear momentum, and impulse.

• Ability to apply results to real-world road safety situations.

Two ice skaters, initially at rest and facing each other, push each other with their hands and move away in opposite directions. The diagram shows the situation before and after the push.

1. a) What is the total linear momentum of the system formed by the two skaters before pushing each other?
2. b) Given that the external forces acting on the system are negligible during the push, what is the total linear momentum afterwards? State the principle on which you base your answer.
3. c) If one skater has twice the mass of the other, which of the two moves off with a higher speed? Reason your answer.

In a game of pool, the cue ball hits another ball that was at rest.

1. a) During the collision, very intense forces act between the balls, while external forces (weight, normal force, and friction with the felt) are comparatively negligible: what quantity is conserved for the system formed by the two balls?
2. b) Is the velocity of each ball necessarily conserved? Explain the difference between conserving the linear momentum of the system and conserving the velocity of each body.

In an amusement park bumper car ride, a bumper car of $2\ \text{kg}$ moving at $3\ \text{m/s}$ hits from behind another car of $4\ \text{kg}$ that is stationary, and they remain locked together after the collision. The diagram shows the situation before and after the collision.

1. a) Calculate the total linear momentum of the system before the collision.
2. b) Applying the conservation of linear momentum, calculate the velocity with which the two locked cars move.
3. c) Check that the total linear momentum after the collision matches the initial momentum.

Two bumper cars collide head-on: one of $3\ \text{kg}$ moving at $4\ \text{m/s}$ and another of $1\ \text{kg}$ coming in the opposite direction, also at $4\ \text{m/s}$. After the collision, they remain locked together.

1. a) Taking the direction of the more massive car as positive, calculate the total linear momentum before the collision (be careful with the signs).
2. b) Calculate the velocity of the combined system after the collision and state in which direction they move.
3. c) Explain what would have happened if the total linear momentum before the collision had been zero.

A tanker car of $20000\ \text{kg}$ moving at $4\ \text{m/s}$ along a straight track collides with two coupled cars, of $10000\ \text{kg}$ each, that are stationary, and the whole assembly becomes coupled together after the maneuver.

1. a) Calculate the linear momentum of the system before the collision.
2. b) Calculate the velocity of the three cars combined after coupling.
3. c) Justify, by stating the corresponding principle and the conditions satisfied, why the conservation of linear momentum can be applied even though the collision deforms the couplings.

In a ballistics lab, a bullet of $0.02\ \text{kg}$ is fired horizontally at $400\ \text{m/s}$ into a wooden block of $1.98\ \text{kg}$ at rest on a horizontal surface. The bullet becomes embedded in the block and the combined system slides until coming to rest. The coefficient of friction between the block and the surface is $\mu = 0.4$ (take $g = 10\ \text{m/s}^2$). The diagram shows the instant before impact and the system sliding afterward.

1. a) Apply the conservation of linear momentum to the collision and calculate the velocity of the bullet-block system immediately after impact.
2. b) Calculate the deceleration of the system (remember that $F_r = \mu \times N$).
3. c) Calculate the distance traveled by the system before coming to rest (use $v^2 = 2 \times a \times x$).

A skater of $60\ \text{kg}$, at rest on the ice, throws a backpack of $3\ \text{kg}$ horizontally at $10\ \text{m/s}$.

1. a) What is the total linear momentum of the skater-backpack system before the throw?
2. b) Applying the conservation of linear momentum, calculate the recoil velocity of the skater.
3. c) Relate this phenomenon to rocket propulsion.

An astronaut of $80\ \text{kg}$ (including spacesuit) is at rest next to her spacecraft, floating in space, and throws a tool of $2\ \text{kg}$ away from herself at $8\ \text{m/s}$.

1. a) What is the total linear momentum of the system before the throw?
2. b) Calculate the recoil velocity of the astronaut.
3. c) How can she use this same principle to return to the spacecraft if she is drifting? Propose a maneuver and justify it.

A firework rocket of $5\ \text{kg}$ is moving horizontally at $8\ \text{m/s}$ when it explodes into two fragments. A fragment of $3\ \text{kg}$ continues in the same direction at $12\ \text{m/s}$. The diagram shows the rocket before exploding and the two fragments after.

1. a) Calculate the linear momentum of the rocket before the explosion.
2. b) Applying the conservation of linear momentum, calculate the velocity of the other fragment, of $2\ \text{kg}$.
3. c) Now suppose that the $3\ \text{kg}$ fragment were to move off at $20\ \text{m/s}$: calculate again the velocity of the $2\ \text{kg}$ fragment and interpret the sign of the result.

A worker of $60\ \text{kg}$ cleaning plastics from a lagoon is standing in a boat of $120\ \text{kg}$, both at rest on the water. Suddenly, he jumps horizontally toward the dock with a speed of $3\ \text{m/s}$.

1. a) Applying the conservation of linear momentum, calculate the recoil velocity of the boat.
2. b) What is the total linear momentum of the worker-boat system before and after the jump?
3. c) Explain why the conservation of linear momentum is no longer useful for predicting the motion if the worker pushes off the dock to jump: is the worker-boat system still an isolated system?

Two laboratory carts, of $1\ \text{kg}$ and $3\ \text{kg}$, are at rest and in contact, with a compressed spring between them. Upon releasing the spring, the cart of $1\ \text{kg}$ moves off at $6\ \text{m/s}$. The diagram shows the carts before and after releasing the spring.

1. a) Applying the conservation of linear momentum, calculate the velocity with which the cart of $3\ \text{kg}$ moves off.
2. b) Check that the total linear momentum of the system after the "explosion" is still zero.
3. c) If the spring is compressed further and the cart of $3\ \text{kg}$ moves off at $3\ \text{m/s}$, at what velocity does the cart of $1\ \text{kg}$ move off?
4. d) Based on the results, explain why, in an explosion, the lighter fragment always moves off faster than the heavier one.