A 15 kg block is firmly attached to a very light horizontal spring (k = 550 N/m) and resting on a smooth horizontal table. Suddenly it is struck horizontally by a 3 kg stone travelling with a velocity of 8 m/s to the right, whereupon the stone rebounds at 2 m/s to the left. Find the maximum distance that the block will compress the spring after the collision.
Before the collision only the stone has momentum. After the collision both the stone and the block have momentum, but the total momentum should not change. Let’s use the law of momentum conservation and calculate the velocity of the block just after the collision.
m1v1i + m2v2i = m1v1f + m2v2f
Where m1 is the mass of the block, m2 is the mass of the stone, v1i and v1f are velocities of the block before and after collision respectively, v2i and v2f are velocities of the stone before and after collision respectively.
15 * 0 + 3 * 8 = 15 * v1f – 3 * 2
v1f = 2 m/s
Next we apply the law of conservation of energy
Ek = Usp
where Ek – the kinetic energy of the block and Usp – the potential energy of the spring.
m1 * v1f2 / 2 = k * x2 / 2
15 * 22 = 550 * x2
x ≈ 0.33 m