Question

An object with m₁ = 5kg is attached to a spring of negligible mass. This mass/spring combination is then slid horizontally on a frictionless surface with a velocity of 5m/s towards a stationary object with m₂ = 6kg. Upon impact, the spring compresses, then we examine two cases. First, find the velocities of the two objects assuming the spring completely relaxes again after the interaction. Second, assume that m₂, after they separate, slides up a frictionless incline.

(a) What is the relative speed of the masses when the spring is maximally compressed?

(b) When the spring is completely compressed, how much potential energy does it have?

(c) If the spring constant is k = 2000 N/m, by how much (what distance) is the spring compressed at that point?

(d) If the spring relaxes while still on at ground, what is the kinetic energy of m₂ just before it slides up the hill?

(e) If there is no friction on the hill, how high up the hill does m₂ go?

(f) If there is friction on the hill, and m₂ goes up the hill to a height of 0.5 meters, how much energy was dissipated?

Answer 1

(a) Initial speed of mass is

Inital speed of mass   is (at rest)

When the masses come in to contact with the spring between them, speed of decreases and speed of increases. At a certain instant both the masses will have the same speed in the same direction. This is the instant when the spring has maximum compression.

Let the positions of masses be and at a certain instant. Hence length of spring is . When spring is compressed maximally, Hence

That is . Hence moment of maximum compression is the moment of equal velocities.

Hence relative speed of masses is zero, when the spring is compressed maximally.

(b) When the spring is completely compressed, potential energy of the spring = loss in kinetic energy of the system of masses.

Conserving momentum of system

When the spring is compressed maximally, . Fom above equation

Initial kinetic energy of system

Initial potential energy of system

Final kinetic energy of the system (at the instant of maximal compression)

Final potential energy of the system,

(c)

Spring is compressed to a maximum distance of

(d) When the masses separate finally, there will be no energy in the spring as it is relaxed. Since there is no loss of energy,the interaction of masses is like an elastic collision.

From conservation of momentum

From conservation of kinetic energy

Dividing the above equations,

Hence final kinetic energy of , just before it slides up a hill is

(e)

If mass goes to a height on the hill,

(f)

If mass goes to a height on the hill with friction,

Where is energy dissipated due to friction.