Momentum is the quantity of motion that an object has.

Momentum helps us understand how much motion something has; if asked you how much it would hurt if you got hit by a ping pong ball, you’d ask, “how fast is it travelling?”, or if someone asked you how much it would hurt if you got hit by something moving at 1 m/s, you’d ask, “how big is the object?”.  Momentum allows us to easily answer questions like these and quantify the motion of an object.  The formula for momentum is p = mv, where p is momentum, m is mass, and v is velocity.

A concept that is often helpful is solving problems regarding momentum is impulse.  Impulse can be quantified by the formula J = Ft, where J is impulse, F is force, and t is time.  Impulse is the change in an object’s momentum when it is acted upon by a force (F) over a time (t).  A very important relationship between impulse and momentum is the impulse-momentum theorem, which states that Ft = pf - pi (impulse = change in momentum).


Example Problem #1

A baseball (.5kg) is travelling with a speed of 15 m/s towards a batter, who hits a ball so that it travels back towards the pitcher with a speed of 20 m/s.  If the ball was in contact with the bat for .012 seconds, how much force was applied to the ball by the bat?

Answer: F = 1,458.33N

Explanation: Using the impulse-momentum theorem, we can say that, since Ft = pf - pi, F(.012) = .5(20-(-15)).  Solving for F, we get that F = 1,458.33N

Conservation of Linear Momentum

We can often use the conservation of linear momentum to help us solve difficult problems regarding collisions.  The formula that we use to solve these is:

mAvAi + mBvBi = mAvAf + mBvBf

There are multiple different types of collisions; there are perfectly inelastic collisions, inelastic collisions, and elastic collisions.  Perfectly inelastic collisions are when the two objects that collide stick to each other after the collision and can be treated as one entity.  Inelastic collisions are collisions where kinetic energy is not conserved (if the objects stick together, then the collision is perfectly inelastic.  Lastly, elastic collisions are those where kinetic energy is conserved.

If there are angles involved in the collision, then the conservation of linear momentum still applies, but you must analyze the collision on both the x-axis and y-axis independently (split motion into components).


Example Problem #2

Two carts (ma = 1kg and mb = 2kg) are travelling towards one another with velocities of 5 m/s.  If cart A is travelling to the right, cart B is travelling to the left, and the collision is perfectly inelastic, what is the velocity of the carts after the collision?

Answer: -5/3 m/s or 5/3 m/s to the left.

Explanation: We can rewrite the equation for the conservation of linear momentum so that it represents a perfectly inelastic collision.  Since the objects stick together after impact, they will have the same final speed.

mAvAi + mBvBi = Vf(mA + mB)

We can plug in 1kg for ma, 2kg for mb, 5 m/s for mAi and -5 m/s for mBi.  Once we have done so, we are left with:

5 + (-10) = Vf(3)

Solving for Vf, we get that Vf = -5/3 m/s or 5/3 m/s to the left.