Momentum

What is momentum?
Momentum of a body is defined as the mass multiplied by the velocity of this object.
Momentum= m x v

Momentum and Newton’s second law of motion:
The resultant force is proportional to the change in momentum per a second.
We know that force = mass x acceleration. So F (mv-mu)/t F m (v-u)/t = ma so F=kma
Momentum is a vector quantity:
Momentum has a direction as well as a magnitude

Momentum and Newton’s first law of motion:
An object remains at rest or in uniform motion unless acted upon by a force.
If an object had a constant momentum, it will have a constant amount of force needed to that will mean that no resultant force acting on it. So it will have a constant velocity unless the mass changes.
Momentum key points

Unit of momentum:
Kgms-1
Symbol of momentum:
P
But what is momentum as a physical quantity?
Momentum is the measure of how much force is needed to stop the moving object or change its velocity (speed or direction)

Momentum is found in lots of examples from our everyday lives. To understand what momentum is we look at two colliding objects. Each object is moving with a certain velocity and has a certain mass. To stop this object a certain force must be applied to counter the velocity. Each unit of mass will have the same velocity so the total force needed will be the velocity multiplied by the mass. This is momentum.
Momentum and acceleration ΔV Δt
If we look at the relation between momentum and Newton’s second law, Δmv Δt
We can replace “a” in F= ma by which will give us F= Δmv Δt
When the mass of the object is constant we can say that:
F=
Impulse
V Δm Δt
The impulse of a force is the force multiplied by the time for which the force acts. The impulse is also equal to the change in momentum because,
F=
Δm Δt
Where the mass is changing and the rate of this change = .
Because the mass is changing at a constant rate V Δm = Δ (m V)
So the change in momentum (Δm V) = F Δt = impulse
Force- time graph mv- mu Δt
A force time graph shows the variation of force acting on an object during a certain time. If the change in the force causes a change in the velocity we can have this equation:
F =
Which is rearranged into Ft = mv – mu, so the force multiplied by the time is equal to the change in momentum. The force multiplied by the time is equal to the area under the graph. This means that the area under the graph is equal to the change in momentum.
Impact
Impact can cause a ball to gain momentum or will cause a change in the balls original momentum. This graph shows the force against the time for a tennis ball rebounding. The constant line on the top of the graph shows the moment of impact.
_
+
Velocity = +u
Velocity = -v

The impact force for this ball the change in momentum is equal to
(-mV) - (mu) (-mv) – (mu) Δt
_
+
Velocity = +u
Velocity = -v
Which means that the impact force
F=

(-2mu) Δt
Also note that if you don’t have any loss in the velocity the impact force will be Because (v=u)

Newton’s third law of motion
Newton’s third law of motion tells us that for every action there is a reaction equal and opposite in direction. Sometimes the reaction force is not as obvious as the action force. For example we know that all objects with a mass act down towards the earth, but the earth also pushes back at those objects.

Conservation of momentum
Momentum (like force) is not lost, it is only conserved. So when 2 objects interact the momentum will transfer from one object to another. This means that the momentum before the interaction is equal to the momentum after the interaction.
If we consider a moving object and a stationary object interacting, the velocity of the moving object and its mass will determine its momentum. When the momentum is transferred to the stationary object, the moving object will still have some momentum but the stationary object has part of it which means it will start moving. So we end up with two objects moving at a lower velocity than the original velocity.
To show this we can use the following example:

The impact force F1 on ball A from ball B changes the velocity of A from UA to UB.
Therefore:

“t” is the contact time between the two balls.
The impact Force F2 on ball B due to ball A changes the velocity of B from UB to VB
Therefore:

Because the two forces are equal and opposite: F2=-F1
This also gives:
=

This can be simplified to: mBVB - mBUB = - (mAVA - mAUA)
This could be rearranged to: mBVB + mAVA= mBUB - mAUA
This is the same as
Total final momentum = total initial momentum
Head on collisions
If two objects collided together they will usually change each other’s momentum. But if the two objects had the same exact momentum and where were travelling in perfectly opposite direction, both objects will stop moving after they collide.
If we have these two identical cars travelling at the same speed (2000kg, 25 ms-1) the initial momentum of both cars is the same (50000 kgms-1) but one of them will be negative so the total momentum will add up to zero so the cars will remain in the position they were at during the collision (sticking to each other).

Elastic and Inelastic collisions
A bouncy ball will rebound when dropped. The height to which it rebounds is equal to the height from which it’s thrown. However a wooden box if dropped will not rebound at all. A plastic ball will rebound but it will not reach the height it was dropped from it.

mB mA VB
VA

* The bouncy ball represents elastic collision in which no momentum is lost during the collision. This menas that the ball will have the same speed (its mass and momentum doesn’t change so the velocity wont either according to M= v x m). this means that energy will be conserved. So the ball will be able to return to its original poisition (due to the tranfer of energy from kinetic to gravitational energy).

* As for the wooden block, it will loose a large portion of energy during the impact which means that it will not have enough energy to enable it to return to the position it was in so it will reamin in the position it was in during the collision. This example is similar to the head on collision in cars explained earlier.

* The plastic ball, on the other hand, will loose only a small amount of energy when it collides with the floor. This means that it will have some energy left to allow it to bounce back. However this energy will not be sufficent for it to bounce back to the same hieght it wazs dropped from, which means that it will rebound at a lower hieght.

To be able to distinguish between collisions, we need to caculate the kinetic energy before and after the collision.

Explosions
An explosion happens when two objects fly apart, each carrying equal and opposite momentum. Perhaps the word explosion brings to mind loud bangs, fire and , but explosions can happen at smaller scale.
If we take for example two woden trolleys, one with a spring loaded bolt, and stick them next to each other. When the spring in the bolt is released we will notice that each car has gone off in a different direction. If the conditions for both trolleys is the same (e.g. mass, volume, material, surface of travel) then the trolley’s will have the same speed.
We can test this:
The total initial momentum = 0
The total momentum after the explosion: mA VA + mB VB using the principle of momentum: mA VA + mB VB=0 mB VB = - mA VA
This means that the two masses will move apart in opposite directions.
We notice for this example that the total kinetic energy of the two trolleys just as they move apart is equal to the potential energy stored in the spring that pushes them apart. However in an explosion involving more objects some of the energy will be lost as sound, heat, and light.
Explosions are very helpful in helping us determine the mass of an object relatively depending on its speed. This could be helpful in observing explosions that happen in outer space.

Closer look at momentum:
Perhaps the most practical use of momentum in every day life is on a pool table. When you use the cue stick to hit one of the balls, you transfer kinetic energy form the stick to the ball. This gives the ball a resultant force causing it to accelerate and have a velocity. This means that it will have momentum. When this ball hits another, the collision will be ealstic meaning that all the energy and the momentum will get transfered to the second ball. The frist pool ball will go backwards, while the second one will now have enough energy to move .

answers:
1 D
2 D
3 B
4 D
5 B
Another example of momentum appliacations is in rugby players. The same principle applies, but sometimes different players have different speeds, masses, and shapes which all effect their momentum.This means that some players are harder to stop than others.

Test your momentum knoweldge: