Momentum calculations

In a nutshell 

All moving objects have momentum, a larger momentum means that it's harder to bring the object to rest. The momentum of an object is the product of its mass and velocity. Momentum is a vector quantity, the direction that the object is travelling in is the direction of the momentum. 

Equations

Variable definitions

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Definition of momentum

The momentum of an object is the mass of the object multiplied by the velocity of the object:

​$$momentum = mass \times velocity$$

​$$p = m \times v$$​​

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Example

If a car with a mass of $$1200 \space kg$$​ is moving with a velocity of $$5 \space m/s$$, what is the momentum of the car?

Begin by writing out the quantities you have been given:

$$m = 1200 \space kg$$​​

$$v = 5 \space m/s$$

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Next, write down the equation you need to use:

$$p=m\times v$$​​

Then, substitute these values into the equation:

​$$p = 1200 \times 5$$​

Don't forget to include your units:

$$6000\, kg\,m/s$$​​

The momentum of a $$1200\,kg$$ car travelling at $$5\,m/s$$​ is $$\underline{6000 \space kg \space m/s}$$.

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Conservation of momentum

A closed system is one in which there are no external force acting. In a closed system. The momentum before an event is equal to the momentum after the event. 

This principle is called the conservation of momentum.

This is the reason that cannons move backwards after they fire a cannonball. The cannonball and cannon are not moving initially, so the total momentum before the cannonball is fired is zero. Once the cannonball is fired, the ball moves forward with a very high speed. Momentum, which is a vector quantity, needs to be conserved so the cannon moves backwards - in the opposite direction to the ball. 

Change in momentum

If a force is applied to a moving object, the momentum of the object must change. The force is proportional to the rate of change of the momentum:

​$$force = \dfrac{change \ in \ momentum}{time} = \dfrac{mass \space \times \space change \space in \space velocity}{time}$$

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​$$F = \dfrac{\Delta p}{t} = \dfrac{m \space \Delta v }{t}$$​​

This is Newton's second law of motion. For a constant mass, the rate of change in velocity is the acceleration of the object and so the equation goes to:   

​$$F=ma$$