text: an unknown mass falls 4 meters than hits a spring with constant 1000 n/m and compresses 1 meter, what is mass? the mass falls four meters than hits the spring which compresses another meter before coming to rest (temporarily)    friction and air resistance can be ignored. we will assume that air and spring friction is negligible. we will also assume that the mass starts falling with a velocity of zero (it isn't thrown, for example).  first, we find the velocity of the mass at the time it strikes the spring. distance = (1/2)gt^2 so: 4 = (1/2)(9.8)t^2, t = 0.90 sec. v = gt = (9.8)(0.90) = 8.85 meters/sec  so the kinetic energy of the mass when it hits is (1/2)mv^2, which is (1/2)m(8.85)^2 or 39.2m joules.  the work done by gravity as the spring compresses is f*d or (9.8)m(1) or 9.8m joules.  next, we need to know the work done by the spring as it compresses. this is the integral of force over distance or int(1000ldl). integrating gives 500 l^2 evaluated from 1 to 0 which is -500 joules.  finally, we have our equation. the kinetic energy of the mass is absorbed by the spring, as is the work done by gravity. so we have :  39.2m + 9.8m = 500 49m = 500 m = 10.2 kg.