Question

a block is moving on a frictionless horizontal surface with velocity |V|=v when it hits a spring. Suppose the spring constant is k. How much is the maximal compression of the spring?

Please help me :)

Answer 1

v = horizontal speed
m = mass of block
k = spring constant
x = maximal compression of the spring

In this case, I think it would be easiest if we looked at energy. Since the block is moving, it has some kinetic energy, given by (1/2)mv^2. When the block hits the spring, it will transfer all this kinetic energy to the spring in the form of elastic potential energy, given by (1/2)kx^2. This transfer will occur until the block comes to a stop, meaning ALL its original kinetic energy will become elastic potential energy. Therefore:

\[\frac{ 1 }{ 2 }mv^2 = \frac{1}{2}kx^2\]\[mv^2 = kx^2\]\[x^2 = \frac{mv^2}{k}\]\[x = \sqrt{\frac{mv^2}{k}}\]

And that's your maximal compression! If you have any questions please ask!