Q7 - What is the force constant, k, of the spring in N/m?

Question

A 4.8*10⁵ kg locomotive is brought to a halt from a speed of 0.75 m/s in 0.45 m by a large spring bumper at the end of its track.

Answer 1

mv²/2 = kd²/2; k = mv²/d² = 4.8·10⁵kg·0.75²m²/s²/(0.45²m²) = 1.33·10⁶N/m

Answer 2

Mechanical energy H is conserved before and after the locomotive comes to a stop.

1. before: only kinetic energy of the locomotive, K = 1/2 mv^2, v = velocity = 0.75 m/s
2. after: only elastic energy of the spring, E_elastic = 1/2 kx^2, x = compression = 0.45m

By equating them, you get: 1/2mv^2 = 1/2 k x^2, which you can solve for the spring constant k = mv^2/x^2.

As a sanity check, this formula has the right units: kg * m^2/s^2 /m^2 = N/m