At a waterpark, sleds with riders are sent along a slippery, horizontal surface

Question

At a waterpark, sleds with riders are sent along a slippery, horizontal surface by the release of a large, compressed spring. The spring with a force constant k = 4500 N/m 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 67.0 kg are pushed against the other end, compressing the spring 0.385 m . The sled is then released with zero initial velocity.

What is the sled's speed when the spring returns to its uncompressed length?

What is the sled's speed when the spring is still compressed 0.225 m ?

Explanation / Answer

KE = PE

1/2mv^2 = 1/2kx^2

mv^2= kx^2

v= sqrt(k/m)*x = sqrt(4500/67)*0.385 = 3.16 m/s

KE = PE

1/2mv^2 = 1/2kx^2

KE = PE

PEi = KEf + PEf

1/2kxi^2 = 1/2mvf^2 + 1/2kxf^2

½*4500*0.385^2 = ½*67*vf^2 + ½*4500*0.225^2    => vf= 2.56m/s

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