Momentum is the product of mass and velocity.
p = mv
p = momentum (kg⋅m/s)
m = mass (kg)
v = velocity (m/s)
Law of Conservation of Momentum
- The total momentum of a system is always the same.
- The total momentum before and after a collision is the same.
Perfectly Inelastic Collision
In this type of collision, two objects collide and then stick together. After the collision, they move with the same velocity. For the purposes of solving math problems, they act like a single object with one velocity and one total mass after the collision.
Momentum is conserved in inelastic collisions, but kinetic energy is not conserved.
Perfectly Elastic Collision
In an elastic collision, two objects bounce off of each other with different velocities at the end of the collision. No kinetic energy is lost in a perfectly elastic equation. In other words, the total kinetic energy before the collision equals the total kinetic energy after the collision. Momentum is also conserved, so there are two separate equations that can be combined to predict the results of a perfectly elastic equation.
Collisions between atoms are perfectly elastic. Other than that, real life collisions are not perfectly elastic – some kinetic energy is “lost,” meaning some of that energy is converted to other forms of energy or transferred elsewhere.
Partially Elastic Collision
Partially elastic collisions are somewhat bouncy, but not as bouncy as perfectly elastic collisions. Some amount of the initial total kinetic energy is lost from the objects, but they still bounce apart.
Calculating Total Momentum of a System
A “system” means a collection of some objects. Each object has its own mass and velocity.
Calculate the momentum of each object separately, then add those values together.
Momentums can have opposite signs if their velocities are pointed in opposite directions. Be sure to include those signs when you add the values together!
Example: calculate the total momentum of the system that includes the blue and red carts below:
- The 12 kg blue cart is moving south at 5.0 m/s
- The 7.0 kg red cart is moving north at 2.0 m/s
For this example, we’ll say that north is positive and south is negative. The sign is part of the velocity.
Blue Cart: p = mv = (12 kg)(−5.0 m/s) = −60 kgm/s
Red Cart: p = mv = (7.0 kg)(+2.0 m/s) = +14 kgm/s
Total momentum = (−60 kgm/s) + (+14 kgm/s) = −46 kgm/s
The total momentum is negative (south) since the southbound card started with more momentum.
Momentum Transfer
In a collision between two carts (or any type of object), one cart gives some of its momentum to the other cart.
Example Numbers:
Before Collision
1 kg blue cart is at rest (momentum = 0)
3 kg red cart is moving 4 m/s (momentum = 12 kgm/s)
After Inelastic (Hit and Stick) Collision
1 kg blue cart is moving 3 m/s (momentum = 3 kgm/s)
3 kg red cart is moving 3 m/s (momentum = 9 kgm/s)
Notice:
1) Total momentum adds up to 12 kgm/s before and after collision
2) Blue cart’s momentum change is +3 kgm/s (it gained momentum)
Δp = 3 – 0 = +3
3) Red cart’s momentum change is -3 kgm/s (it lost momentum)
Δp = 9 – 12 = -3
4) 3 kgm/s of momentum was transferred from the red cart to the blue cart.
5) Total amount of transferred momentum is zero: +3 + -3 = 0
Δpred + Δpblue = 0
6) For a perfect two object collision, the amount of momentum change has to be the same for each object but with opposite signs, (This also means the momentum change of each cart is equal in magnitude but in opposite directions)