Question

A railroad wagon (mass m₁) has a velocity v₁ . It collides with a wagon (mass m₂) which is initially at rest. Both wagons roll without friction after the collision. The second wagon is connected via a spring (spring constant k) with a block (mass m₃) that lies on a rough surface (coefficient of static friction µ₀). Assume the impact to be plastic and determine the maximum value of v₁ so that the block stays at rest.

Answer 1

The total momentum of the system remains unchanged during the impact. This yields the velocity ¯v of the wagons after the collision : 

 We assume that the block stays at rest. Then we can write down the equilibrium condition

The force of static friction H and the normal force N have to satisfy the condition of static friction:

H ≤ µ₀N → F ≤ µ₀m₃g .

With

F = kx → x ≤ µ₀m₃g/k

we obtain the maximum compression of the spring: 

x_max = µ₀m₃g/k

Now we apply the Conservation of Energy Law to determine v₁max: