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P5 X) Momentum – Part 2
P5 X) Momentum – Part 2
We can work out the momentum of an object by using the formula below.
In the above formula, p is momentum measured in kilogram metres per second (kg m/s), m is mass measured in kilograms (kg) and v is velocity measured in metres per second (m/s).
Momentum in a Closed System
A closed system means that there are no external forces (e.g. friction or air resistance). In a closed system, the total momentum before an event is equal to the total momentum after an event. This is called the law of conservation of momentum.
A closed system means that there are no external forces (e.g. friction or air resistance). In a closed system, the total momentum before an event is equal to the total momentum after an event. This is called the law of conservation of momentum.
There are two different types of events that we can have; collisions and explosions. Collisions are where two objects are moving towards each other. Explosions are where two objects are moving away from each other.
Collision Example
A collision is where two objects are moving towards each other. We are going to look at a collision between 2 roller skaters; Andrew and Ben. Andrew has a mass of 60 kg and he is travelling towards Ben with a velocity of 2.5 m/s (Andrew is traveling towards the right on the diagram below). Ben has a mass of 40 kg and he is stationary. The diagram below shows Andrew and Ben with their masses and velocities.
A collision is where two objects are moving towards each other. We are going to look at a collision between 2 roller skaters; Andrew and Ben. Andrew has a mass of 60 kg and he is travelling towards Ben with a velocity of 2.5 m/s (Andrew is traveling towards the right on the diagram below). Ben has a mass of 40 kg and he is stationary. The diagram below shows Andrew and Ben with their masses and velocities.
During the collision, Andrew and Ben grab onto each other and travel off together towards the right. Find the velocity of both of the skaters after the collision. Assume that the collision is happening in a closed system.
According to the law of conservation of momentum, the momentum before a collision will be equal to the momentum after a collision. The momentum before the collision will be the sum of the momentums of Andrew and Ben separately.
According to the law of conservation of momentum, the momentum before a collision will be equal to the momentum after a collision. The momentum before the collision will be the sum of the momentums of Andrew and Ben separately.
We now find the momentum of Andrew (PA) and Ben (PB) seperately. We find momentum by multiplying the mass by the velocity. Andrew has a mass of 60 kg and a velocity of 2.5 m/s. Ben has a mass of 40 kg and is stationary, so he has a velocity of 0 m/s. We sub these values in to find the momentum before the collision.
The momentum before the collision is 150 kg m/s.
Due to the law of conservation of momentum, the momentum after the collision will be equal to the momentum before the collision; the momentum after the collision will also be 150 kg m/s.
Due to the law of conservation of momentum, the momentum after the collision will be equal to the momentum before the collision; the momentum after the collision will also be 150 kg m/s.
During the collision, Andrew and Ben grab onto each other. This means that we can imagine them as one entity with a combined mass; the combined mass will be 100 kg (Andrew + Ben = 60 kg + 40 kg = 100 kg).
We can now work out the velocity of both of the skaters after the collision.
We can now work out the velocity of both of the skaters after the collision.
We want to find the value of v and not 100v. Therefore, we divide both sides of the equation by 100.
This tells us that the velocity of the two skaters after the collision is 1.5 m/s.
We can now compare the velocities and momentums of the individuals before and after the collision.
We can now compare the velocities and momentums of the individuals before and after the collision.
- Andrew’s velocity has decreased. Before the collision his velocity was 2.5 m/s and after the collision his velocity is 1.5 m/s. Also, as Andrew’s velocity has decreased after the collision, his momentum will also have decreased after the collision.
- Ben’s velocity has increased. Before the collision his velocity was 0 m/s (he was stationary) and after the collision his velocity is 1.5 m/s. Also, as Ben’s velocity has increased after the collision, his momentum will also have increased after the collision.
Explosion Example
Explosions cause objects to move away from each other in opposite directions. Before an explosion takes place, the objects are stationary. An example of an explosion is a bullet being shot out of a gun. Another example is an individual jumping off of a boat. The law of conservation of momentum applies to explosions as well as collisions. Here is the law of conservation of momentum.
Explosions cause objects to move away from each other in opposite directions. Before an explosion takes place, the objects are stationary. An example of an explosion is a bullet being shot out of a gun. Another example is an individual jumping off of a boat. The law of conservation of momentum applies to explosions as well as collisions. Here is the law of conservation of momentum.
We are now going to have a question to do with the shooting of a bullet.
Question
A bullet has a mass of 40 grams and a gun has a mass of 800 grams. The bullet and the gun are stationary. The operator of the gun pulls the trigger, which results in the bullet being fired out at a velocity of 600 m/s. Work out the velocity of the recoil of the gun (the recoil of a gun is the gun moving backwards when a bullet is fired).
Before the trigger on the gun was pulled, both the gun and the bullet are stationary. This means that the momentum before the explosion is zero.
Question
A bullet has a mass of 40 grams and a gun has a mass of 800 grams. The bullet and the gun are stationary. The operator of the gun pulls the trigger, which results in the bullet being fired out at a velocity of 600 m/s. Work out the velocity of the recoil of the gun (the recoil of a gun is the gun moving backwards when a bullet is fired).
Before the trigger on the gun was pulled, both the gun and the bullet are stationary. This means that the momentum before the explosion is zero.
Due to the law of conservation of momentum, the momentum before the explosion will be equal to the momentum after the explosion. As the momentum before the explosion is zero, the momentum after the explosion will also be zero.
I am now going to draw a diagram of what happens when the trigger is pulled (after the explosion). For my diagram below, when the trigger is pulled, the bullet travels towards the right and the gun travels in the opposite direction towards the left (this is the recoil). I have also added the masses of the gun and the bullet to the diagram.
Velocity is a vector quantity, which means that it has magnitude (size) and direction. Therefore, we need to have one of the directions (right or left) as a positive velocity and the other direction as a negative velocity. I am going to have the right as positive and the left as negative. Also, the masses of the items for momentum need to be in kilograms. These are the values for the gun and bullet:
We can now sub these values into the formula.
- The gun has a mass of 800 grams (0.8 kg) and an unknown velocity, which I will let be v.
- The bullet has a mass of 40 grams (0.04 kg) and a velocity of 600 m/s (the velocity is positive as it is going towards the right).
We can now sub these values into the formula.
We now solve the equation to find v in the same way that we would solve a maths equation. The first step is to get the v’s on their own, which we do by moving the + 24 from the right side of the equation to the left. We are able to do this by doing the opposite, so we take 24 from both sides of the equation.
We want to find the value of v and not 0.8v, so we divide both sides of the equation by 0.8.
This tells us that the velocity of the gun is -30 m/s. A negative value for velocity means that the gun is moving in a leftwards direction (the opposite direction to the direction of the bullet). This means that the velocity of the recoil of the gun is 30 m/s to the left (the opposite direction of the bullet).
We can now compare the velocities and momentums. The recoil of the gun is 30 m/s to the left and the velocity of the bullet is 600 m/s to the right. The velocity of the recoil of the gun is lower than the velocity of the bullet because the gun has a larger mass than the bullet (the mass of the gun is 0.8 kg which is larger than the mass of the bullet which is 0.04 kg). The momentum of the gun and the bullet are equal and in opposite directions.
We can now compare the velocities and momentums. The recoil of the gun is 30 m/s to the left and the velocity of the bullet is 600 m/s to the right. The velocity of the recoil of the gun is lower than the velocity of the bullet because the gun has a larger mass than the bullet (the mass of the gun is 0.8 kg which is larger than the mass of the bullet which is 0.04 kg). The momentum of the gun and the bullet are equal and in opposite directions.
A Word Question
Sometimes in the exam, you will be asked worded questions about the law of conservation of momentum rather than numerical questions. If this is the case, you will need to state what the law of conservation of momentum is, and then compare the momentums and velocities both before and after the event (collision or explosion).
Sometimes in the exam, you will be asked worded questions about the law of conservation of momentum rather than numerical questions. If this is the case, you will need to state what the law of conservation of momentum is, and then compare the momentums and velocities both before and after the event (collision or explosion).