1. PES 1000 – Physics in Everyday Life
Momentum And Impulse
PES 1000 – Physics in Everyday Life

2. Momentum, Impulse, and Motion
Forces
We know from experience that forces can create motion, represented by velocity
Newton’s Laws explain motion in the following way:
Forces cause acceleration (Newton’s second law)
Acceleration changes velocity
Another useful approach to explain motion uses the concepts of momentum and impulse
In this approach, forces generate impulse
Impulse can change the amount of momentum of an object
The momentum is related to its velocity
Some situations may be easier to understand using the momentum and impulse method
The momentum and impulse method can be derived from Newton’s Laws and vice versa.
Forces cause acceleration (F=ma)
Forces generate an impulse
Acceleration changes velocity
Impulse changes momentum
Motion (velocity)

3. Definitions and equations
Momentum
It is the product of mass and velocity, so it is a vector quantity
The variable used is often 𝑝
So 𝑝 =𝑚∗ 𝑣
Its units are kg*m/s
A large momentum can come from a large mass moving slowly or a small mass moving quickly.
Impulse
It is a change in momentum, ∆ 𝑝
The variable used is often 𝐼
It is the cumulative effect of force acting over a time interval, so 𝐼 = 𝐹 ∗∆𝑡
You must take direction into account
It units are N*s, which is equivalent to kg*m/s (momentum units)

4. If force is constant
Adding up the forces acting at each instant of time can be complicated if the force is changing magnitude and direction throughout the time interval.
In the special case of a constant force, however, it is easy to do:
𝐼 = 𝐹 ∗𝑡 = 𝐹 ∗∆𝑡 , where ∆𝑡 is the entire time interval
An example of this situation is one we’ve seen before: a ball thrown into the air
Ignoring drag, there is a constant force on the ball, its weight 𝐹 𝐺
The initial momentum of the ball is its upward velocity times its mass, 𝑝 𝑖 =𝑚∗ 𝑣 𝑖
The impulse due to gravity acting during the upward motion, 𝐹 𝐺 ∗𝑡, is opposite the ball’s motion, so it reduces the initial momentum until it becomes zero at the top of the flight.
Gravitational force acting during the ball’s descent gradually increases the ball’s momentum, now in the downward direction, until it reaches your hand with a downward momentum of 𝑝 𝑓 =𝑚∗ 𝑣 𝑓
momentum FG

5. If Force Varies
A more complicated example is if the force varies in magnitude, but not direction, during some time interval, Dt
If we plot the force at each instant, it might look something like this:
The total interaction time is the time interval Dt
There is a maximum force during the interval, Fmax.
Since impulse is force * time, the area under the curve is equal to the impulse
Imagine a rectangle with the same time interval and also the same area. The height of this rectangle is the average force, Favg. F t
Fmax
Favg Dt

6. Example: Car colliding with a wall
The car is traveling at some speed, so it has momentum m*v as it approaches the wall.
As the car’s front begins to touch the wall, the interaction interval begins.
The force begins rising over the (short) time interval, reaching a peak at some point.
The force acting opposite the car’s motion over the time interval reduces the car’s momentum to zero.
Eventually the force of the wall drops to zero at the end of the interval.
Even though the force graph may be complicated, we still know the area under the graph; it is the impulse, which is the change in momentum of the car. Since the car comes to rest, the change in momentum is just the total initial momentum.
Replacing the complicated graph with a rectangle of the same area, we can characterize the average force on the car, which is large enough to bend metal. F t
Favg Dt

7. Car/Wall collision with crash barrels
We’d like to reduce these forces.
The total impulse must be the same as before. It is the initial momentum of the car.
The only option to reduce the force is to increase the interaction time interval.
If the bottom of the rectangle (the time interval) gets wider, the height (average force) must get smaller to have the same area.
If the average force is reduced, the maximum force will also be reduced, perhaps to safe levels for the passengers.
In general, the longer the time interval of the interaction, the lower the forces involved. F t Dt
Favg
Fmax F t Dt
Favg
Fmax

8. Other examples of increased time, decreased force
Helmet
If your head hits the ground, its momentum before it hits the ground will be reduced to zero. The short collision time will require a large, potentially damaging, force. Wearing a helmet with a compression zone increases the total interaction time, thus reducing the force required to reduce your head’s momentum.
Brain
When your head stops suddenly, your brain, because it has inertia, and moves forward until it is stopped by the skull. The longer the time to stop your brain, the lower the average forces on it. So your body has a cushion of cerebro-spinal fluid between the brain and the skull to reduce the forces this way.
Bungee
Two equal-sized guys jump off a bridge, one with a rope tied to him, one with a bungee cord. When they reach the end of their respective ropes, they will have the same speed, and therefore the same momentum. The rope and bungee will eventually stop them, so their stopping impulse is the same. The rope has a very short stopping time, so it has a huge stopping force. The bungee has a long interaction time, so the forces are very much smaller.

9. Conclusion
The approach using momentum and impulse is another way to examine motion.
Momentum is a vector and is computed by mass * velocity.
Impulse is a change in momentum and is computed by force * time.
During a collision, momentum is removed by impulse (force * time interval).
If time interval is short, forces must be large.
If time interval is longer, the forces will be lower.
This is the physics behind many safety mechanisms.