At a waterpark, sleds with riders are sent along a slippery,...

At a waterpark, sleds with riders are sent along a slippery, horizontal surface by the release of a large compressed spring. The spring with force constant $k=40.0 \mathrm{N} / \mathrm{cm}$ and negligible mass rests on the frictionless horizontal surface. One end is in contact with a stationary wall. A sled and rider with total mass 70.0 $\mathrm{kg}$ are pushed against the other end, compressing the spring 0.375 $\mathrm{m}$ . The sled is then released with zero initial velocity. What is the sled's speed when the spring (a) returns to its uncompressed length and (b) is still compressed 0.200 $\mathrm{m} ?$

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Key Concepts

Conservation of Energy

This principle states that in an isolated system with no non-conservative forces like friction, the total mechanical energy remains constant. It is used to equate the potential energy stored in a compressed spring to the kinetic energy of an object when the spring is released.

Elastic Potential Energy

Elastic potential energy is the energy stored in a deformed object, such as a compressed or stretched spring, and is calculated using the formula U = ½ k x². This concept is crucial for determining how much energy is available to be converted into kinetic energy when the spring returns to its equilibrium state.

Kinetic Energy

Kinetic energy, given by the expression KE = ½ m v², is the energy associated with the motion of an object. In the context of this problem, the conversion of elastic potential energy to kinetic energy allows us to calculate the speed of the object after the spring has been released.

Hooke's Law

Hooke's Law describes the linear relationship between the force exerted by a spring and its displacement from equilibrium (F = -kx). This law underpins the behavior of the spring and the calculation of the elastic potential energy stored during compression.

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