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M Momentum and Impulse The momentum of a particle of mass moving with velocity , is . It is a vector quantity like velocity and is measured in Newton-seconds Newton-seconds (N s). Example One Find the magnitude of the momentum possessed by a car of 750kg moving with a speed of 25 . mome moment ntum um
When dealing with momentum it is very important that you have the correct units, remember that mass is in kg and velocity is in . 750 25 18,750 Ns Ns
If the velocity of a body changes from to , then logically we must assume that the momentum of the body will change. Then change in momentum can be quantified (measured) by finding the initial momentum, and the final momentum, . Example Two Find the change in momentum of a body of mass 4kg when its speed changes from:
I. II.
4 to 7 in the same direction. 6 to 2 in the opposite direction.
When dealing with questions like this it always useful to sketch diagrams to illustrate the mathematical concept. I.
In this example we consider movement to the right to be positive, although it’s not important in this example it will in the next.
init initiial mom momentu entum m 4 4 16 N s
In this second example we will also consider the movement to the right to be positive and movement to the left will be negative.
init initiial momen omentu tum m 4 6 24 N s
initially
finally
6 −1
2 −1
4
4
4 −2 −2 −8 N s
The momentum changes from 24Ns to -8Ns which is therefore a change of 32Ns.
Impulse The impulse of a force is defined as length of time a force acts (i.e. impulse ). We also know that , therefore we can assume that; impulse
In our previous tutorial on uniform acceleration we are told that; −
We can substitute this into our equation for impulse and we get; imp impulse ulse − − chan change ge in mome moment ntum um
Therefore; impulse of a force change in momentum
Conservation of Momentum In the previous tutorial on connected particles we came across Newton’s Third Law of Motion. When a collision occurs between two particles A and B the force exerted by A on B is equal and opposite to the force exerted by B on A. Without any other external forces not only are the forces equal and opposite but the momentum of A and B are also equal in magnitude and opposite in direction.
When these two bodies collide the gain in momentum of one body will be accompanied by a decline in momentum of the other body. Therefore the sum of the momentum of A and B before a collision will be equal to the sum of the momentum after the collision. This is essentially the principle of conservation of (linear) momentum . As with the previous examples a diagram is often essential when attempting questions involving the collision of particles. You should usually draw a diagram representing the 2
M condition before the collision and then draw a second diagram after the collision has happened. Remember that since momentum is a vector quantity, include positive and negative signs, otherwise you will get an incorrect answer. When a collision occurs usually one of three things occurs; 1. The two bodies coalesce (join together) in this instance the two bodies become one and move at a common velocity. 2. The two bodies move in the same direction but at different velocities and as individual particles. 3. One of the bodies rebounds and travels in the opposite direction. Example Three A body of mass 5kg moving on a smooth horizontal surface at 4 , collides with a second body of 3kg which is at rest. After the collision the bodies coalesce. Find the common speed of the bodies after impact. before collision
4 5
−1
after collision
−1 at rest
8
3
If we assume that the velocities to the right are positive we get the following momentum before the collision; initial initial moment momentum um 5 4 3 0 20 N s inal momentum 8 8 N s
By considering the principle of conservation of momentum; 20 8
20 8
2.5 2.5
Therefore the speed of the coalesce body after the collision is 2.5 .
M Example Four A body of mass 5kg, is moving with a velocity of 9 , when it collides c ollides with a a body of mass 3kg, travelling in the the same direction at a velocity of 2 , immediately after the collision the body of mass 5kg is travelling in the same direction but at a velocity of 6 , given that the direction the 3kg body is unchanged how fast is it travelling?
As with all questions dealing with momentum lets draw a diagram of the bodies immediately before and after their collision. after collision
before collision
9 −1
2 −1
6 −1
5
3
5
3
If we take the velocities to the right to be positive; initial initial momentum momentum 9 5 2 3 4 5 6 initia initiall mome momentu ntum m 51 N s
Now that we have our initial momentum we can look towards finding the final velocity of the 3kg mass by forming an equation for . 6 5 3 30 3 30 3 51 3 51 − 30 21 7
In the final example we will look at what happens when to bodies collide and immediately after the collision begin to travel in different directions.
M Example Five Two bodies collide as shown in the diagram below on a smooth horizontal surface. Find the speed of immediately after the impact. before collision
after collision
7 −1
2 −1
2 −1
2
3
2
3
In this example as with previous examples we will take velocities to the right as being positive.
First let’s take a look at what happens prior to the collision in terms of momentum; initial initial momentu momentum m 7 2 −2 3 initial mo momentum 14 − 6 8 N s
Now let’s take a look at what happens after the collision; −2 2 3 −4 3
We can now apply our principle of conservation of momentum to find the value of the velocity . 3 − 4 8 3 12 4
When you do lots of practice questions you will come to see that momentum is a pretty straight forward topic, examiners are a mean bunch however and try to make it more difficult by perhaps adding a third collision. The important thing to do is to keep your cool and take it one collision at a time. They may also ask you to give the final velocity in terms of a certain symbol (usually ) so it is always worth reading up on re-arranging equations. equations.
