Elastic Collision Formula

An elastic collision is a collision where both kinetic energy, KE, and momentum, p, are conserved. This means that KE₀ = KE_f and p_o = p_f. Recalling that KE = 1/2 mv², we write 1/2 m₁(v_1i)² + 1/2 m₂(v_i)² = 1/2 m₁(v_1f)² + 1/2 m₂ (v_2f)², the final total KE of the two bodies is the same as the initial total KE of the two bodies. And, since p = linear momentum = mv, then we write m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f.

[A] m₁v_1i + m_{2v2i = m1v1f + m2v2f}

[B] 1/2 m₁(v_1i)² + 1/2 m₂(v_i)² = 1/2 m₁(v_1f)² + 1/2 m₂ (v_2f)²

KE = kinetic energy

p = momentum

m = mass, kg

m_i = mass of 1st object

m₂= mass of 2nd object

v = velocity, m/s

v₁ = velocity of 1st object

v₂ = velocity of 2nd object

v_i = initial velocity

v_f = final velocity

Elastic Collision Formula Questions:

1) A red ball of mass 0.2 kg hits a blue ball of mass 0.25 kg, in an elastic collision, and the red ball comes to a stop. The red ball has a velocity of 5 m/s, and the blue ball was at rest. What is the final velocity of the blue ball?

Answer: The mass of the 1st ball, m₁ = 0.2 kg; the mass of the 2nd ball, m₂ = 0.25kg. The initial velocity of the 1st ball, v_1i = 5 m/s; the initial velocity of the 2nd ball, v_2i = 0; the final velocity of the 1st ball, (v_1f) = 0.

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

(0.2 kg)(5 m/s) + (0.25 kg)(0 m/s) = (0.2 kg)(0) + (0.25 kg)(v_2f)

1.0 kg.m/s + 0 = 0 + (0.25 kg)(v_2f)

1.0 kg.m/s = (0.25 kg)(v_2f)

(1.0 kg.m/s) / 0.25 kg = (v_2f)

4 m/s = (v_2f)

2) Use the equation for conservation of kinetic energy in an elastic collision to determine the final velocity for the blue ball.

Answer: The mass of the 1st ball, m₁ = 0.2 kg; the mass of the 2nd ball, m₂ = 0.20kg. The initial velocity of the 1st ball, v_1i = 5 m/s; the initial velocity of the 2nd ball, v_2i = 0; the final velocity of the 1st ball, (v_1f) = 0.

1/2 m₁(v_1i)² + 1/2 m₂(v_i)² = 1/2 m₁(v_1f)² + 1/2 m₂ (v_2f)²

1/2 (0.2 kg)(5m/s)² + 1/2 (0.2 kg)(0) = 1/2 (0.2 kg)(0) + 1/2 (0.2 kg)(v_2f)²

1/2 (0.2 kg)(5m/s)² = 1/2 (0.2 kg)(v_2f)²

(5m/s)² = (v_2f)²

25 m²/s² = (v_2f)²

v_2f = √25 m²/s²

v_2f = 5 m/s