Two Object Problems

When you’re given a two-object problem and you’re asked to calculate the change in momentum, take into account the initial momentum and the final momentum individually first of each object. pinitial = pfinal. When you solve for the final momentum, decide whether you should use an individual mass value or the sum of all masses in the system. 

If you’re solving for the final velocity of the system, then you should absolutely use the sum of both masses regardless of how many objects there are. If an object is initially at rest, its initial momentum will be zero since its velocity in the beginning was zero (p = m*v).  

Change in momentum is proportional to the force and time so if you decrease T or F, you decrease the change in momentum. 

Image Credit: ponderisd.net

Example Problem #1:

A tennis ball of mass 0.05 kg is hit with a racket, causing it to travel at a velocity of 40 m/s. The racket has a mass of 0.5 kg and is traveling at a velocity of 10 m/s. What is the total momentum of the ball and the racket before and after the collision?

Solution:

Before the collision, the momentum of the ball is given by the formula: momentum = mass * velocity

The mass of the ball is 0.05 kg, and its velocity is 40 m/s.

Therefore, the momentum of the ball before the collision is: momentum = 0.05 kg * 40 m/s = 2 kg*m/s

Before the collision, the momentum of the racket is given by the formula: momentum = mass * velocity

The mass of the racket is 0.5 kg, and its velocity is 10 m/s.

Therefore, the momentum of the racket before the collision is: momentum = 0.5 kg * 10 m/s = 5 kg*m/s

Before the collision, the total momentum of the ball and the racket is: 2 kgm/s + 5 kgm/s = 7 kg*m/s

After the collision, the total momentum of the ball and the racket remains constant, since there are no external forces acting on the system.

Therefore, the total momentum of the ball and the racket after the collision is also: 7 kg*m/s

Example Problem #2:

Two cars are traveling on a highway. Car A has a mass of 2000 kg and is traveling at a velocity of 60 m/s. Car B has a mass of 1000 kg and is traveling at a velocity of 40 m/s. The cars collide head-on and stick together. What is the total momentum of the cars before and after the collision?

Solution:

Before the collision, the momentum of car A is given by the formula: momentum = mass * velocity

The mass of car A is 2000 kg, and its velocity is 60 m/s.

Therefore, the momentum of car A before the collision is: momentum = 2000 kg * 60 m/s = 120000 kg*m/s

Before the collision, the momentum of car B is given by the formula: momentum = mass * velocity

The mass of car B is 1000 kg, and its velocity is 40 m/s.

Therefore, the momentum of car B before the collision is: momentum = 1000 kg * 40 m/s = 40000 kg*m/s

Before the collision, the total momentum of the cars is: 120000 kgm/s + 40000 kgm/s = 160000 kg*m

After the collision, the combined mass of the cars is 2000 kg + 1000 kg = 3000 kg.

The combined velocity of the cars is not given, so we will assume it to be v.

The total momentum of the cars after the collision is therefore: momentum = 3000 kg * v m/s

Since the total momentum of the cars before and after the collision must be equal, we can set these two expressions equal to each other:

120000 kgm/s + 40000 kgm/s = 3000 kg * v m/s

160000 kg*m/s = 3000 kg * v m/s

v = 160000 kg*m/s / 3000 kg = 53.333 m/s

Therefore, the combined velocity of the cars after the collision is 53.333 m/s.

Example Problem #3:

A ball of mass 0.1 kg is thrown with a velocity of 10 m/s. A catcher catches the ball with a velocity of 5 m/s. What is the change in momentum of the ball?

Solution:

The change in momentum of the ball is given by the formula: change in momentum = mass * change in velocity

The mass of the ball is 0.1 kg, and its change in velocity is: 10 m/s - 5 m/s = 5 m/s

Therefore, the change in momentum of the ball is: change in momentum = 0.1 kg * 5 m/s = 0.5 kg*m/s

Example Problem #4:

A rocket of mass 100 kg is launched from the surface of the Earth with a velocity of 100 m/s. The rocket carries a payload of 50 kg. What is the total momentum of the rocket and the payload before and after the launch?

Solution:

Before the launch, the momentum of the rocket is given by the formula: momentum = mass * velocity

The mass of the rocket is 100 kg, and its velocity is 0 m/s (since it is stationary before the launch).

Therefore, the momentum of the rocket before the launch is: momentum = 100 kg * 0 m/s = 0 kg*m/s

Before the launch, the momentum of the payload is also zero, since its velocity is zero.

Before the launch, the total momentum of the rocket and the payload is: 0 kgm/s + 0 kgm/s = 0 kg*m/s

After the launch, the momentum of the rocket is given by the formula: momentum = mass * velocity

The mass of the rocket is 100 kg, and its velocity is 100 m/s.

After the launch, the momentum of the rocket is: momentum = 100 kg * 100 m/s = 10000 kg*m/s

After the launch, the momentum of the payload is given by the formula: momentum = mass * velocity

The mass of the payload is 50 kg, and its velocity is 100 m/s (since it is carried along by the rocket).

Therefore, the momentum of the payload after the launch is: momentum = 50 kg * 100 m/s = 5000 kg*m/s

After the launch, the total momentum of the rocket and the payload is: 10000 kgm/s + 5000 kgm/s = 15000 kg*m/s

Example Problem #5:

A baseball of mass 0.15 kg is hit with a bat, causing it to travel at a velocity of 50 m/s. The bat has a mass of 1 kg and is traveling at a velocity of -20 m/s (i.e. in the opposite direction). What is the total momentum of the ball and the bat before and after the collision?

Solution:

Before the collision, the momentum of the ball is given by the formula: momentum = mass * velocity

The mass of the ball is 0.15 kg, and its velocity is 50 m/s.

Therefore, the momentum of the ball before the collision is: momentum = 0.15 kg * 50 m/s = 7.5 kg*m/s

Before the collision, the momentum of the bat is given by the formula: momentum = mass * velocity

The mass of the bat is 1 kg, and its velocity is -20 m/s.

Therefore, the momentum of the bat before the collision is: momentum = 1 kg * -20 m/s = -20 kg*m/s

Before the collision, the total momentum of the ball and the bat is: 7.5 kgm/s + -20 kgm/s = -12.5 kg*m/s

After the collision, the total momentum of the ball and the bat remains constant, since there are no external forces acting on the system.

Therefore, the total momentum of the ball and the bat after the collision is also: -12.5 kg*m/s

Interpreting Different Scenarios

You need to be able to interpret different scenarios, especially with problems that have multiple external forces acting on the system. When you see problems like that, make sure you find the vector sum of the forces after calculating individual forces. If you’re looking for the momentum, you can either find individual momentums of the objects or find the total momentum of the system. Understand how to apply the formula and make sure you can explain your thought process because some FRQ’s ask for conceptual explanations. 

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