Conservation of Momentum
Brook Edgar & Hannah Shuter
Teachers
Explainer Video
Momentum
Momentum is a property that all moving objects have - it depends on both how heavy the object is and how fast it's going. A lorry moving slowly might have the same momentum as a car moving quickly because the lorry is much heavier.
Formula:
Remember that velocity is a vector (has direction), so momentum also has direction. If something moves to the right, its momentum is positive. If it moves left, its momentum is negative.
Example: A car of mass is moving to the right with a velocity of . The momentum of the object could be calculated using:
However, if this car was moving at the same speed but to the left, the velocity would now be , so the momentum would be .
Objects that are not moving have a momentum of .
Worked Example:
Calculate the momentum of a rock falling at .
Answer:
First, convert mass -> kg by dividing by :
Then:
Worked Example:
An ice skater with a mass of is skating to the left with a velocity of . Calculate the momentum and state the direction.
Answer:
The velocity is to the left so we need to make it negative:
The momentum is to the left as the momentum is always in the same direction as the velocity.
Conservation of Momentum
In a closed system, the total momentum before an event equals the total momentum after the event. An "event" usually means a collision or explosion.
Conservation of Momentum:
A closed system means there are no external forces messing things up - like friction or air resistance. In real life, we often ignore these to make calculations simpler, or we do experiments on air tracks where friction is very small.
Momentum in Explosions
An explosion in Physics is any event where objects go from stationary to moving.
Example: imagine two ice skaters, Girl A and Girl B. If the girls are stationary and then push off each other, they move in opposite directions - this would count as an "explosion". Girl A has a mass of and girl B has a mass of . If Girl B moves to the right at , we can calculate girl A's velocity using the law of conservation of momentum:
Since the girls were stationary to start with, the total momentum before the event would be . Therefore, the total momentum after the event also has to be - this is conservation of momentum. This doesn't mean that the girls won't be moving, it just means that if we add their momentums together it will equal . For this to happen, one of the momentums will have to be negative - this means the girls will move in opposite directions.
Lots of momentum questions use ice skaters as there is minimal friction on ice, therefore it is nearly a closed system.
Momentum in Collisions
Whether objects bounce off each other or stick together, the total momentum is always conserved when objects collide.
Objects Sticking Together:
When two objects collide and stick together (like two cars crashing and becoming tangled), they move off together with a combined velocity.
Example: a car with mass of skids on ice at a velocity of , collides with a stationary car of mass and they stick together and move off. If we assume that this collision happens in a closed system (no external forces so no friction), we can work out the speed at which the cars move using conservation of momentum:
Momentum before the collision:
The car is stationary, so it has a momentum of before the collision.
If we know the momentum before the collision is , then we know the total momentum after the collision is also . Because the cars are stuck together, we can now treat them as "one" larger car of mass :
Objects Bouncing Apart:
Sometimes, when objects collide they rebound from one another and separate after the collision. The calculation is similar but each object has its own final velocity to work out.
Example: On a pool table, the white ball (mass of ) is moving at and collides with the stationary red ball (). The balls collide and move off in the same direction. If we know that the red ball moves at , we can calculate how fast the white ball must be moving:
The total momentum before the collision would just be the momentum of the white ball, as the red ball is stationary:
If we know the total momentum before the collision, we know the total momentum after the collision must be the same, so the sum of the momentum of the white ball and the momentum of the red ball must equal :
Worked Example:
Two cars collide head-on. Car X () is travelling right at before the collision and car Y () is travelling left at . They stick together. Calculate their velocity after collision.
Answer:
First, we need to calculate the total momentum before the collision. Because the cars are travelling in different directions, we need to give one of the velocities a negative sign - convention states that objects travelling to the left are given the negative value:
Now we know the total momentum before the collision, we know the total momentum after the collision will be the same. We are told in the question the cars stick together so we can now treat them as "one" object of mass :
Worked Example:
Two carts on a frictionless track. Cart A () moves right at . Cart B () is stationary. They collide and cart B moves off at . Calculate the final velocity of cart A.
Answer:
First we need to calculate the total momentum before the collision. Because cart B is stationary, the total momentum will be only the momentum of cart A:
If we know the total momentum before the collision, we know the total momentum after the collision must be the same:
Worked Example:
A cannon of mass fires a cannon ball of mass at a velocity of . Calculate the recoil velocity of the cannon.
Answer:
We know that the total momentum before the collision is as neither the cannon or cannonball are moving, so the total momentum after the collision must also be :
So the cannon moves in the opposite direction to the cannonball.
Practice Questions
Two stationary ice skaters push off each other on frictionless ice. Skater A (mass ) moves backwards at . Calculate the velocity of skater B (mass ) after the push.
-> Check out Hannah's video explanation for more help.
Answer:
A ball moving at to the right collides head-on with a ball moving at to the left. They stick together. Calculate their velocity after the collision.
-> Check out Hannah's video explanation for more help.
Answer: