Collision of Mass Spring System

A mass m₁, with initial velocity V₀, strikes a mass-spring system m₂, initially at rest but able to recoil. The spring is massless with spring constant k (see Figure 1.1). There is no friction.

Figure 1.1

a) What is the maximum compression of the spring?

b) If, long after the collision, both objects travel in the same direction, what are the final velocities V₁ and V₂  of m₁ and m₂ respectively?

SOLUTION

a) The maximum compression of the spring occurs at the moment when the velocities of the two masses and become equal (see Figure 1.2). For this moment we can write conservation of momentum and energy as

(1)

(2)

where A is the maximum compression of the spring; from (1)

Figure 1.2

and from (2)

where μ = m₁ m₂/(m₁ + m₂) is the reduced mass.

b) If, long after the collision, both masses move in the same direction, it means m₁ > m₂, that and the spring will not be compressed. So we have

We can easily find V₁ and V₂ from these equations

where γ = m₂/m₁.