Elastic Collision

• Newton’s cradle: It is a device that demonstrates elastic collision with swinging spheres.
• Rutherford scattering: The scattering between the alpha particles and the gold nuclei is elastic.
• Gravitational slingshot: Also known as a gravity assist, it uses the gravitational force of a planet to change the course of an approaching satellite.

In an elastic collision:

• Total kinetic energy before and after the collision remains unchanged
• Total momentum before and after the collision remains unchanged
• Energy is not dissipated through heat or friction
• Colliding objects do not deform

Suppose two objects, A and B, with masses m_A and m_B traveling with velocities v_Ai and v_Bi collide head-on. After the collision, the objects change direction and velocity. Let v_Af and v_Bf be the final velocities, which we are interested in calculating.

Applying the law of conservation of momentum,

Applying the law of conservation of kinetic energy,

As there are two equations and two unknowns, we use the method of simultaneous equations to solve for v_Af and v_Bf.

These equations give the final velocities of objects A and B after a head-on collision. Let us look at a few specific cases.

Case 1: Object A is at rest, and object B is in motion.

Here, v_Ai = 0. Therefore,

Case 2: Object B is at rest, and object A is in motion.

Here, v_Bi = 0. Therefore,

Case 3: Objects A and B have the same mass.

Here, m_A = m_B = m

According to the equations, the two objects bounce off each other, exchanging velocities. So, if two objects of equal masses collide head-on, they exchange momentum.

Now, suppose B is at rest (v_Bi = 0). Then,

It means that A comes to a stop, and B picks up the same speed as A.

Problem 1: Two bumper cars, A and B, at an amusement park collide elastically as one approaches the other directly from the rear. Car A, in front, has a mass of 650 kg. Car B, behind, has a mass of 550 kg. The car in front was traveling at 4.30 m/s while the car behind hit him with a velocity of 5.70 m/s.  What are their final velocities after the collision?

Solution:

Given,

m_A = 650 kg, m_B = 550 kg, v_Ai = 4.3 m/s, and v_Bi = 5.7 m/s

We have,

v_Af = (m_A – m_B)/(m_A + m_B) v_Ai + 2m_B/(m_A + m_B) v_Bi

v_Af = (650 kg – 550 kg)/(650 kg + 550 kg) x 4.3 m/s + 2 x 550 kg/(650 kg + 550 kg) x 5.7 m/s

Or, v_Af = 5.58 m/s

And,

v_Bf = 2m_A/(m_A + m_B) v_Ai – (m_A – m_B)/(m_A + m_B) v_Bi

Or, v_Bf = 2 x 650 kg/(650 kg + 550 kg) x 4.3 m/s – (650 kg – 550 kg)/(650 kg + 550 kg) x 5.7 m/s

Or, v_Bf = 4.18 m/s

Problem 2: Calculate the velocities of two objects, A and B, following an elastic collision, given that m_A = 1.75 kg, m_B = 3.25 kg, v_A = 6.00 m/s, and v_B = 0.

Solution:

Given,

m_A = 1.75 kg, m_B = 3.25 kg, v_A = 6.00 m/s, and v_B = 0.

Here, object A is moving, and object B is at rest (v_Bi = 0). Therefore,

v_Af = (m_A – m_B)/(m_A + m_B) v_Ai

Or, v_Af = (1.75 kg – 3.25 kg)/(1.75kg + 3.25 kg) x 6 m/s

Or, v_Af = – 1.8 m/s

And,

v_Bf = 2m_A/(m_A + m_B) v_Ai

Or, v_Bf = 2 x 1.75 kg/(1.75 kg + 3.25 kg) x 6 m/s

Or, v_Bf = 4.2 m/s

The negative sign indicates that object A bounces and move in the opposite direction after the collision.