Momentum

For two objects colliding in 1-dimension.

Before Collision:

Basic Principles

Conservation of Momentum: The total momentum of the system (both objects) is the same before and after the collision.

Momentum is: $$ \text{Momentum} = M = m \cdot v $$ where:

m = mass
v = velocity

The total momentum before collision is: $$ \begin{align*} M_{before} &= m_1 \cdot v_1 + m_2 \cdot v_2 \\ &= 4 \cdot 2 + 6 \cdot (-3) \\ &= 8 + (-18) \\ &= -10 \; \text{kg m/s} \end{align*} $$
Write a programs for:

Inelastic Collision: If the two objects stick together after the collision, what is their final velocity.
Elastic Collision: Velocity of the first object (m₁) after the collision, if you are given the velocity of the second object after the collision, and the masses of both objects.

The underlying principle is that momentum is conserved after the collision, so the total momentum before collision is equal to the total momentum after collision.

Notes and Code	Outcome
Inelastic Collision In inelastic collisions the two objects stick together after they collide. Therefore, after collision they can be considered a single mass. After Collision The total momentum *after collision (M_a* ) is equal to the momentum before collision (M_b ): $$ \begin{align} M_{a} &= M_{b} \\ &= -10 \; \text{kg m/s} \end{align} $$ Therefore after collision**: $$ \begin{align} M_{a} &= m \cdot v_a \\ -10 &= 10 \cdot v_a \\ \frac{-10}{10} &= v \\ v &= -1 \; \text{m/s} \end{align} $$	To write a program to calculate the final velocity, you can calculate values for each step, or you can do the algebra to determine a single equation for this problem.
Elastic Collision In elastic collisions the objects bounce off of each other after they collide. Determine the velocity of object 1 after the collision if the velocity of object 2 is -2 m/s after the collision.	To write a program to calculate the velocity of object 1 after collision, if you are given the starting velocities and masses, and the velocity of object 2 after the collision.

Notes and Code

Outcome

Inelastic Collision

In inelastic collisions the two objects stick together after they collide. Therefore, after collision they can be considered a single mass.

After Collision

The total momentum after collision (M_a ) is equal to the momentum before collision (M_b ): $$ \begin{align} M_{a} &= M_{b} \\ &= -10 \; \text{kg m/s} \end{align} $$ Therefore after collision: $$ \begin{align} M_{a} &= m \cdot v_a \\ -10 &= 10 \cdot v_a \\ \frac{-10}{10} &= v \\ v &= -1 \; \text{m/s} \end{align} $$

To write a program to calculate the final velocity, you can calculate values for each step, or you can do the algebra to determine a single equation for this problem.

Elastic Collision

In elastic collisions the objects bounce off of each other after they collide.

Determine the velocity of object 1 after the collision if the velocity of object 2 is -2 m/s after the collision.

To write a program to calculate the velocity of object 1 after collision, if you are given the starting velocities and masses, and the velocity of object 2 after the collision.