U3AOS1 Topic 12: Collisions

Collision questions are almost like momentum questions, but now there's just an extra step which is classifying the collision as Elastic or Inelastic .

The difference between Elastic and Inelastic collisions is very simple and is shown in the table below.

Elastic	Inelastic
An elastic collision is when KE is conserved	An inelastic collision is when KE is NOT conserved
$ KE_{i} = KE_{f} $	$ KE_{i} \neq KE_{f} $

If energy is always conserved then why won't the initial and final KEs be the same?

It's because a part of the energy can get dissipated from the collision in other forms of energy such as heat, sound, etc.

Therefore the overall net energy in the entire system is still conserved, while the net Kinetic Energy of the system is different.

Solving collision questions

When solving collision questions the first thing generally is to calculate the speeds using the momentum equations.

From there, you calculate the initial and final kinetic energies.

\ [KE_{i} = KE_{f} \]

\ [ \frac{1}{2}m_1(u_1)^2+\frac{1}{2}m_2(u_2)^2 = \frac{1}{2}m_1(v_1)^2 +\frac{1}{2}m_2(v_2)^2 \]

(this is assuming there are only two objects if there are more add more m, u and v values)

SOLVING COLLISIONS FULLY:

1. Calculate all speed and mass variables using momentum equations

2. Substitute all variables into kinetic energy equations to calculate the initial and final KE values

3. Determine whether the collision is elastic or inelastic

Created with GeoGebra®, by Tom Walsh, Link

Use the calculator below to help with the calculations:

Example 1

If Cart A is travelling at $ 5 $ m/s West and Cart B is travelling $ 1 $ m/s East toward each other until they crash and move together at $ 3 $ m/s West. Assume both carts have a mass of $ 50 $kg. Is the collision elastic or inelastic?

Solution:

The question is asking to determine whether the collision is elastic or not.

Therefore, the initial and final energies must be checked.

If they are equal then the collision is elastic.

We are given that both masses travel at the same velocity after the collision.

Step 1: Label the direction so that West is negative and the East is positive

\[ \leftarrow is -, \rightarrow is + \]

Step 2: Write down all known variables

\[ m_{1} = m_{2} = 50 \text{kg}, u_{1} = -5 \text{m/s}, u_2 = 1 \text{m/s}, u_3 = -3 \text{m/s} \]

Step 3: Choose the appropriate equation

\[ \frac{1}{2} m_{1} (u_{1})^2 + \frac{1}{2} m_{2} (u_{2})^2 = \frac{1}{2} m_{1+2} (u_{3})^2 \]

Step 4: Substitute values

\[ \frac{1}{2}(50)(-5)^2 + \frac{1}{2}(50)(1)^2 = \frac{1}{2}(50+50)*(-3)^2 \]

Step 5: Solve carefully

\[ \frac{1}{2}(50)(25) + \frac{1}{2}(50)(1) = \frac{1}{2}(100)*(9) \]

\[ (25)(25) + (25) = 50(9) \]

\[ 625 + 25 \neq 450 \]

The collision is inelastic because the initial and final energy are not equal.

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