Elastic & Inelastic Collisions (Cambridge (CIE) A Level Physics): Revision Note

Two similar spheres, each of mass m and velocity v are travelling towards each other.

The spheres have a head-on elastic collision. What is the total kinetic energy after the impact?

Answer:

Step 1: Equate the kinetic energy before and after the collision

• In an elastic collision, the kinetic energy of the system is conserved

$E_{kbefore}=E_{kafter}$

Step 2: Write an expression for the kinetic energy before the collision

$E_{kbefore}=\frac{1}{2}m{v}^2+\frac{1}{2}m{v}^2$

$E_{kbefore}=m{v}^2=E_{kafter}$

• Therefore, the correct answer is C

Despite velocity being a vector, kinetic energy is never negative because v² is always positive.

Two trolleys X and Y are of equal mass. Trolley X moves towards trolley Y which is initially stationary.

After the collision, the trolleys join and move together.

Prove that this collision is inelastic.

Answer:

Step 1: Write an expression for the kinetic energy of the system before the collision

$E_{kbefore}=\frac{1}{2}{m}_x{v}_x^2+0$

• Object Y is stationary before the collision, so its kinetic energy is zero

Step 2: Write an expression for the kinetic energy of the system after the collision

$E_{kafter}=\frac{1}{2}\left(m_x+m_y\right)v_{x+y}^2$

• Both trolleys are of equal mass, therefore m_x + m_y = 2m

$E_{kafter}=\frac{1}{2}\left(2m\right)v_{x+y}^2$

Step 3: Use conservation of momentum to find the speed after the collision

$m{v}_x=2m{v}_{x+y}$

$v_{x+y}=\frac{v_x}{2}$

Step 4: Compare the expressions and determine if they are equal

$E_{kafter}=\frac{1}{2}\left(2m\right){\left(\frac{v_x}{2}\right)}^2=\frac{1}{4}m{v}_x^2$

$E_{kbefore}=\frac{1}{2}m{v}_x^2$

$E_{kafter}=\frac{1}{2}{E}_{kbefore}$

$E_{kbefore}\ne E_{kafter}$

• The kinetic energy before the collision is not equal to the kinetic energy after the collision

• Therefore, the collision is inelastic

Although kinetic energy may not always be conserved, remember momentum will always be conserved.