Momentum

Let’s talk about what momentum is. Linear momentum is defined as the product of the mass m and the velocity v of an object which in equation form is p=m*v. The change in momentum of an object is a vector in the direction of the net force that's being exerted on the object. The units are kg * m/s for linear momentum.

It’s important to note that a constant linear momentum p is the momentum of an object of mass m that is moving in a straight line with a velocity v. Remember that the velocity is a vector and so is the momentum whereas the mass is a scalar. Momentum can also be applied to objects that are rotating or that have a torque acting upon them. If questions are referring to rotating objects, then the problems will specify “angular momentum,” not linear.  

When you solve for linear momentum in any question, be consistent with the direction you choose to be negative. For example, if you choose to make downwards negative, keep that direction negative throughout the problem. Be sure to consider the direction of the velocity. 

To find the total momentum, you should find the momentum of all directions and then find the sum of the momentum. Total Momentum = p1 + p2 + p3 ..., etc. 

Here are some key things to remember about momentum:

• Momentum is a measure of an object's motion. It is the product of an object's mass and velocity.
• Momentum is a vector quantity, meaning it has both magnitude and direction.
• The momentum of an object can be calculated using the formula: Momentum = mass * velocity
• The law of conservation of momentum states that the total momentum of a closed system remains constant, unless there is an external force acting on the system.
• The conservation of momentum is a useful tool for solving problems involving collisions, explosions, and other interactions between objects. Example Problem 1:

A car is traveling at a speed of 30 m/s when it slams on the brakes and comes to a stop in 5 seconds. What is the momentum of the car before and after it stopped?

Solution:

The momentum of the car before it stopped is given by the formula: momentum = mass * velocity

The mass of the car is not given, so we will assume it to be 1000 kg.

The momentum of the car before it stopped is: momentum = 1000 kg * 30 m/s = 30000 kg*m/s

The momentum of the car after it stopped is zero, since its velocity is zero.

Therefore, the change in momentum of the car is: 30000 kgm/s - 0 kgm/s = 30000 kg*m/s

Example Problem 2:

A ball of mass 0.2 kg is thrown with a velocity of 20 m/s. What is the momentum of the ball?

Solution:

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

The mass of the ball is 0.2 kg, and its velocity is 20 m/s.

Therefore, the momentum of the ball is: momentum = 0.2 kg * 20 m/s = 4 kg*m/s

Example Problem 3:

A car of mass 2000 kg is traveling at a speed of 10 m/s when it collides with a stationary truck of mass 3000 kg. After the collision, the car and the truck moved together with a velocity of 5 m/s. What is the momentum of the car and the truck before and after the collision?

Solution:

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

The mass of the car is 2000 kg, and its velocity is 10 m/s.

Therefore, the momentum of the car before the collision is: momentum = 2000 kg * 10 m/s = 20000 kg*m/s

Before the collision, the momentum of the truck is zero, since its velocity is zero.

After the collision, the combined mass of the car and the truck is 2000 kg + 3000 kg = 5000 kg.

After the collision, the combined velocity of the car and the truck is 5 m/s.

The momentum of the car and the truck after the collision is: momentum = 5000 kg * 5 m/s = 25000 kg*m/s

The change in momentum of the car and the truck after the collision is: 25000 kgm/s - 0 kgm/s = 25000 kg*m/s

Impulse

Newton’s 2nd Law was applied to derive this formula: Sum of the Forces=change in momentum/change in time or F=p/t. This means that the net force acting on an object is equal to the change in its momentum divided by the elapsed time. This equation will remain valid regardless of the changes in mass. Another way to denote this relationship is to say that p = Pfinal - Pinitial which equals P=mFvFinal -mInitialvInitial. So, if the mass remains constant, P=mv.

Rocket thrust refers to (change in mass/change in time) * velocity. The mass of a rocket changes because its engines release fuel as they are fired up. IF fuel is expelled with the speed v and at the rate change in mass/ change in time, the rocket experiences a thrust force. 

Impulse is defined as the average force exerted by an object times the amount of total time elapsed. Impulse is represented by J = Average Force * time. The impulse can be found by finding the area underneath a force vs. time graph. The units for impulse are kgm/s or Ns, same as momentum. Impulse is a vector and points in the same direction as the average force. 

The Momentum Impulse Theorem says that J=p = Ft. Thus, if we know the impulse or change in momentum delivered on an object and the time interval, we can find the average force that caused the impulse. Always consider the direction of the forces and momentum. The forces that act upon an object affects its change in momentum. Make sure you know how to read graphs that relate to impulse, force, and momentum. 

Image Credit: physicsforum.com

The area of this graph will give you the impulse. 

Example Problem 1:

A bowling ball of mass 6 kg is rolling down the lane with a velocity of 10 m/s when it collides with the pins. The collision lasts for 0.2 seconds, and the ball comes to a stop after the collision. What is the impulse of the collision?

Solution:

The impulse of the collision is given by the formula: impulse = force * time

The force of the collision is given by the formula: force = mass * acceleration

The mass of the ball is 6 kg, and its acceleration during the collision is given by the formula: acceleration = change in velocity/time

The change in velocity of the ball during the collision is: 10 m/s - 0 m/s = 10 m/s

The time of the collision is 0.2 seconds.

Therefore, the acceleration of the ball during the collision is: acceleration = 10 m/s / 0.2 s = 50 m/s^2

The force of the collision is therefore: force = 6 kg * 50 m/s^2 = 300 N

The impulse of the collision is therefore: impulse = 300 N * 0.2 s = 60 N*s

Example Problem 2:

A baseball of mass 0.5 kg is thrown with a velocity of 50 m/s. The ball hits a catcher's glove and comes to a stop in 0.01 seconds. What is the impulse of the collision?

Solution:

The impulse of the collision is given by the formula: impulse = force * time

The force of the collision is given by the formula: force = mass * acceleration

The mass of the ball is 0.5 kg, and its acceleration during the collision is given by the formula: acceleration = change in velocity/time

The change in velocity of the ball during the collision is: 50 m/s - 0 m/s = 50 m/s

The time of the collision is 0.01 seconds.

Therefore, the acceleration of the ball during the collision is: acceleration = 50 m/s / 0.01 s = 5000 m/s^2

The force of the collision is therefore: force = 0.5 kg * 5000 m/s^2 = 2500 N

The impulse of the collision is therefore: impulse = 2500 N * 0.01 s = 25 N*s

Example Problem 3:

A golf ball of mass 0.045 kg is struck with a club, causing it to travel at a velocity of 100 m/s. The collision lasts for 0.005 seconds. What is the impulse of the collision?

Solution:

The impulse of the collision is given by the formula: impulse = force * time

The force of the collision is given by the formula: force = mass * acceleration

The mass of the ball is 0.045 kg, and its acceleration during the collision is given by the formula: acceleration = change in velocity/time

The change in velocity of the ball during the collision is: 100 m/s - 0 m/s = 100 m/s

The time of the collision is 0.005 seconds.

Therefore, the acceleration of the ball during the collision is: acceleration = 100 m/s / 0.005 s = 20000 m/s^2

The force of the collision is therefore: force = 0.045 kg * 20000 m/s^2 = 900 N

The impulse of the collision is therefore: impulse = 900 N * 0.005 s = 4.5 N*s

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