Momentum: definition, conservation and calculations
In a nutshell
All objects with mass that are moving 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
definiTION
SYMBOL EQUATION
Momentum
p=mv
Force
F=tmΔv
Variable definitions
QUANTITY NAME
SYMBOL
dervied UNIT
si base UNITs
momentum
p
kgms−1
kgms−1
mass
m
kg
kg
velocity
v
ms−1
ms−1
force
F
N
kgms−2
time
t
s
s
Definition of momentum
The momentum of an object is the mass of the object multiplied by the velocity of the object:
p=mv
Example
If a car with a mass of 1200kg is moving with a velocity of 5ms−1, what is the momentum of the car?
First state the variables from the question:
m=1200kg
v=5ms−1
Next, write down the equation you need to use:
p=mv
Then, substitute these values into the equation:
p=1200×5
Don't forget to include your units:
6000kgms−1
The momentum of a1200kgcar travelling at 5ms−1 is6000kgms−1.
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:
mu=mv
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.
Example
For the diagram below which shows two balls colliding, write a balanced equation for the conservation of momentum.
Firstly, consider the momentum before the collision:
momentumbefore=m1u1+m2u2
From the diagram, u2=0ms−1:
momentumbefore=m1u1
Now, consider the momentum after the collision:
momentumafter=m1v1+m2v2
Therefore as momentumbefore=momentumafter:
m1u1=m1v1+m2v2
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:
F=tΔp=tmΔv
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
Example
A 15g bullet accelerates to 350ms−1 after being fired from a gun with mass 1.5kg. What is the velocity of the recoiling gun?
First state the variables from the question:
mb=1.5×10−2kg,vb=350ms−1,mg=1.5kg
Then state the equation needed:
mgug+mbub=mgvg+mbvb
Rearrange the equation:
0=mgvg+mbvbmgvg=−mbvbvg=−mgmbvb
Sub in and solve:
vg=−1.515×10−3×350vg=−3.5ms−1
The velocity of the recoiling gun is−3.5ms−1.
Note: As the velocity is negative, it means that the gun is travelling in the opposite direction to the bullet.