Momentum & Impulse in 1D | Edexcel A Level Further Maths: Further Mechanics 1 Revision Notes 2017

Impulse-Momentum Principle in 1D

What is momentum?

Any object that has mass and is moving has momentum
Momentum measures the quantity of motion that an object has
The momentum of a particle is defined as the product of its mass ( $m$ kg) and its velocity ( $v m s^{- 1}$ )
- Momentum = mv
The SI unit for momentum is $kg m s^{- 1}$
Momentum is a vector quantity - so it has a magnitude and direction
- The direction of the momentum of a particle is the same as the direction of motion of the particle
- The momentum is negative if the velocity is negative

What is impulse?

Impulse measures the effect of a force acting on a particle over time, it could be thought of as a "push"
If a constant force ( $F N$ ) acts on a particle for $t$ seconds then the impulse ( $I$ ) of the force is defined to be the product of the force and time
- $I = F t$
The SI unit for impulse is N s (newton seconds) which is equivalent to $kg m s^{- 1}$
- This is the same as the units for momentum
Impulse is a vector quantity – so it has magnitude and direction
- The direction of the impulse of a force is the same as the direction of the force

What is the impulse-momentum principle?

The Impulse-Momentum Principle states that impulse is equal to the change in momentum
- $I = m v - m u$
- or equivalently $I = m (v - u)$
- where m is the mass, u is the initial velocity and v is the final velocity

What happens when two objects are in contact?

If two objects are in contact with each other then by Newton’s Third Law there will be equal and opposite reaction forces
This means there will be equal and opposite impulses
For example, consider hitting a tennis ball with a racket, there will be
- an impulse exerted by the racket on the ball which propels the ball forward
- an impulse exerted by the ball on the racket which reduces the velocity of the racket
- The magnitudes of these impulses are equal but they are in opposite directions

ball striking a tennis racket showing the impulse on both in opposite directions

Exam Tip

Always define a positive direction and be careful with negatives. Use common sense to see if your answer makes sense - would you expect the velocity to have increased or decreased?
If two objects collide, remember that the impulse on both objects is the same, just in opposite directions

Worked example

A car with mass 1200 kg is driving to the right along a smooth horizontal road with speed 16 m s^-1 . The driver applied a constant braking force of magnitude 1800 N for 5 seconds.

a) Find the magnitude of the impulse of the braking force.

3-4-1-momentum-_-impulse-example-solution-2_a

b) State the direction of the impulse. 3-4-1-momentum-_-impulse-example-solution-2_b

c) Find the speed of the car 5 seconds after the braking force was applied.

3-4-1-momentum---impulse-example-solution-2-c-fixed

Conservation of Momentum in 1D

What is a direct collision?

A direct collision is when two objects are travelling along the same straight line when they collide
Before the collision:
- One of the objects could be stationary
- The two objects could be travelling in the same direction with the faster object behind the slower one
- The two objects could be travelling in opposite directions towards each other
After the collision:
- One or both of the objects could be stationary
- The two objects could be travelling in the same direction with the faster object in front of the slower one
- The two objects could be travelling in opposite directions away from each other
- The two objects could coalesce (merge to form one object) and travel in either direction
Explosions work like direct collisions and are when an object separates into two objects travelling along the same straight line
- An example of this is a bullet being fired from a gun, the bullet moves forwards and the gun recoils backwards
- For an explosion it is possible that the object is initially stationary and then splits into two objects moving in opposite directions

What is meant by conservation of momentum?

The principle of conservation of momentum states that when two objects collide the total momentum is unchanged
- Total momentum before collision = Total momentum after collision
- This only works if there are no external forces acting on the objects
If an object changes direction after a collision then its velocity changes between positive and negative
- It is important to be clear about which direction is positive
It can be written as: $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$
- One object has mass $m_{1}$ kg, velocity $u_{1} m s^{- 1}$ before the collision and $v_{1} m s^{- 1}$ after the collision
- The other object has mass $m_{2}$ kg, velocity $u_{2} m s^{- 1}$ before the collision and $v_{2} m s^{- 1}$ after the collision

before and after diagrams for two particles colling, showing their velocities and the equation for conservation of momentum

How do I use conservation of momentum to solve collision problems?

STEP 1: Choose the positive direction
STEP 2: Draw a before/after diagram
- Clearly show the mass, speeds and directions
- If a direction is unknown, then choose any direction and if you get a negative value for its velocity it means it is travelling in the opposite direction
- If the two objects coalesce then you can either consider them as two particles moving in the same direction with the same speed or consider them as one particle and add together their masses
STEP 3: Form an equation using the conservation of momentum
- Be careful with negatives
- If an arrow is in the opposite direction to the positive direction, then its velocity is negative
STEP 4: Solve and give answer in context
- You might need to find the speed and/or direction after a collision

Exam Tip

Always draw clear diagrams with arrows!
When considering directions, use common sense - if two particles are travelling in the same direction then they will not collide if the faster one is in front. Two particles can't go through each other (at least not at this level of mathematics…) so if two objects are travelling towards each other, then at least one of them must change direction after the collision.

Worked example

Two particles P and Q, with masses 3 kg and 5 kg respectively, are travelling in opposite directions towards each other along the same straight line on a smooth horizontal table when they collide directly. Immediately before the collision the speeds of P and Q are 4 m s^-1 and 2 m s^-1 respectively. Immediately after the collision the direction of motion of P is reversed and its speed is 1 m s^-1.

Find the speed of Q immediately after the collision, and state whether the direction of motion of Q is changed by the collision.

Momentum & Impulse in 1D (Edexcel A Level Further Maths: Further Mechanics 1)

Revision Note

Impulse-Momentum Principle in 1D

What is momentum?

What is impulse?

What is the impulse-momentum principle?

What happens when two objects are in contact?

Exam Tip

Worked example

Conservation of Momentum in 1D

What is a direct collision?

What is meant by conservation of momentum?

How do I use conservation of momentum to solve collision problems?

Exam Tip

Worked example

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Author: Dan