A runaway railroad car,with mass 36 × 10⁴ kg,coasts across a level track at 2.7 m/s when it collides with a spring-loaded bumper at the end of the track.If the spring constant of the bumper is 2.7 × 10⁶ N/m,what is the maximum compression of the spring during the collision? (Assume the collision is elastic. )

Question

Quizplus · Accepted Answer

Explanation:The kinetic energy of the railroad car is converted to elastic potential energy in the spring. The maximum compression of the spring occurs when all of the kinetic energy has been converted to elastic potential energy. The kinetic energy of the railroad car is:$$KE = \frac{1}{2}mv^2 = \frac{1}{2}(36 	imes 10^4 	ext{ kg})(2.7 	ext{ m/s})^2 = 1.33 	imes 10^6 	ext{ J}$$The elastic potential energy stored in the spring is:$$PE = \frac{1}{2}kx^2$$where k is the spring constant and x is the compression of the spring. Setting the kinetic energy equal to the elastic potential energy, we get:$$1.33 	imes 10^6 	ext{ J} = \frac{1}{2}(2.7 	imes 10^6 	ext{ N/m})x^2$$Solving for x, we get:$$x = \sqrt{\frac{2(1.33 	imes 10^6 	ext{ J})}{2.7 	imes 10^6 	ext{ N/m}}} = 0.99 	ext{ m}$$Therefore, the maximum compression of the spring is 0.99 m.

A Runaway Railroad Car,with Mass 36 × 104 Kg,coasts Across

A Runaway Railroad Car,with Mass 36 × 10⁴ Kg,coasts Across