Question

Two bodies of masses 5 kg and 10kg approach each other with velocities 10 m/s and 5 m/s respectively. Calculate their velocities after collision, assuming the collision to be elastic.​

Answer 1

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Refer to the attachment!

Answer 2

An elastic collision is a collision where both the Kinetic Energy, KE, and momentum, p are conserved. In other words, it means that KE0 = KEf and po = pf.

Let m1 = mass of body 1 = 5 kg

m2 = mass of body 2 = 10 kg

Vi1 = Initial velocity of body 1 = 10 m/s

Vi2 = Initial velocity of body 2 = 5 m/s

Vf1 = Final velocity of body 1

Vf2 = Final velocity of body 2

As momentum is conserved,

m1 × Vi1 + m2 × Vi2 = m2 × Vf2 + m1 × Vf1

5 × 10 + 10 × 5 = 10 × Vf2 + 5 × Vf1

100 = 10*Vf2 + 5*Vf1

20 = 2Vf2 + Vf1

∴ Vf1 = 20 - 2*Vf2           ------ (i)

As Kinetic Energy is conserved,

m1V1i² + m2V2i² = m1V1f² + m2V2f²

5*100 + 10*25 = 5V1f² + 10V2f²

∴ 750 = 5( V1f² + 2*V2f²)

∴ 150 = V1f² + 2*V2f²

Substituting (i),

150 = (20 - 2*Vf2)² + 2*Vf2²

150 = 400 + 4*Vf2² - 80Vf2 + 2*Vf2²

150 = 400 + 6*Vf2² - 80Vf2

6*Vf2² - 80*Vf2 + 250 = 0

Solving the quadratic equation,

We get roots as 8.34 and 5.

Thus, Vf2 = 8.34 and 5 m/s

Substituting in (i)

Vf1 = 3.32 or 10 m/s