A runaway railroad car with mass 3.0 ✕ 105 kg coasts across a level track at 2.0 m/s when it collides elastically with a spring-loaded bumper at the end of the track. If the spring constant of the bumper is 2.0 ✕ 106 N/m, what is the maximum compression of the spring during the collision?

Maximum compression = 77.5cm

Explanation:

We are given that;

mass; m = 3 x 10^ (5) m/s

Speed; v = 2 m/s

Spring Constant; k = 2 x10^ (6) N/m

From conservation of energy. Kinetic energy is transformed into potential energy of spring

Kinetic energy = (½) mv ²

Elastic potential energy of spring = (½) kΔx²

Thus,

(½) mv² = (½) kΔx²

1/2 will cancel out and we have;

mv² = kΔx²

Now we are looking for maximum compression which is Δx, so let's make Δx the subject of the formula;

Δx² = mv²/k

Δx = √ (mv²/k)

Plugging in relevant values;

Δx = √ ((3 x 10^ (5) x 2²) / 2 x10^ (6))

Δx = √ (0.6) = 0.775 m ≈ 77.5 cm