A One-Dimensional Inelastic Collision

Question

Block 1, of mass m1= 9.90 kg, moves along a frictionless air track with speed v1= 23.0 m/s. It collides with block 2, of mass m2= 37.0 kg, which was initially at rest. The blocks stick together after the collision.

Part A: Find the magnitude p_i of the total initial momentum of the two-block system.

Part B: Find v_f the magnitude of the final velocity of the two-block system.

Part C: What is the change ΔK=K_final−K_initial in the two-block system's kinetic energy due to the collision?\

Express your answer numerically

Answer 1

Hi Omar,

We can analyze this inelastic collision by applying the conservation of linear momentum (ΔP = 0).

Conceptual/extra details: Remember that this idea of momentum conservation comes from F = ma in the scenario where the sum of all forces is equal to zero. This is true for our scenario here because when cars collide on a flat frictionless surface they apply equal and opposite forces on each other. However, If this collision were to instead take place on a slanted surface for example, the net force would NOT be zero because gravity would have an influence (gravity not be balanced by the normal force in this case) and momentum would NOT be conserved.

Part A, B

ΔP = P_f - P_i = 0

What is P_initial, the initial momentum of the system? Let's take the direction of block 1s motion to be the positive direction.

P_i = m₁v₁ + m₂v₂ = m₁v₁ + 0 (the second block's initial velocity is zero)

What is P_final? Remember that the blocks are sticking together, and thus acting as one combined mass at the end of the scenario. Together, they move with some final velocity v_f

P_f = (m₁+m₂)v_f

Apply these equations into equation 1

ΔP = P_f - P_i = ((m₁+m₂)v_f) - (m₁v₁) = 0

This gives

(m₁+m₂)v_f = (m₁v₁)

Then just solve this for v_f

Part C

ΔKE = KE_f - KE_i

Initial KE is:

KE_i = (1/2) m₁v₁² + (1/2) m₂v₂² = (1/2) m₁v₁² + 0 (the second block's initial velocity is zero)

final KE is:

KE_f = (1/2) (m₁+m₂)v_f²

Then just plug this into the ΔKE = KE_f - KE_i equation. You will notice that the kinetic energy is actually NOT conserved for this collision. This isn't because energy is beying destroyed. Some of the kinetic energy from the initial state is just simply lost as the blocks collide with each other and interact as they stick (this energy results in the sound of the collision, the heating of the colliding objects, and their deformation).