Final answer:
To find the initial speed of the clay, the conservation of momentum and energy are used. The initial speed is calculated to be approximately 3.433 m/s when the mass of the block, the mass of the clay, the spring constant, and the compression of the spring are given.
Step-by-step explanation:
To find the clay's initial speed v, we will use conservation of momentum and energy conservation principles. Since the clay and block stick together after the collision, this is an inelastic collision. However, the system conserves momentum. When the clay and block compress the spring, mechanical energy (kinetic energy converted to elastic potential energy) is conserved.
The conservation of momentum before and after the collision is given by:
mv + 0 = (m + M)V
where v is the initial speed of the clay, V is the speed of the clay and block after the collision, m is the mass of the clay, and M is the mass of the block.
Next, we use the conservation of energy when they collide with the spring:
½(m + M)V² = ½kx²
where k is the spring constant and x is the compression of the spring. Solving the system of equations resulting from momentum conservation and energy conservation, we get:
V = √x(k / (m + M))
Plugging V back into the momentum conservation equation, we get:
v = (m + M) / m * √x(k / (m + M))
Using M = 2.1 kg, m = 13 g (which is 0.013 kg), k = 136 N/m, and a = 0.05 m for the spring compression, the calculation yields:
v = (2.1 kg + 0.013 kg) / 0.013 kg * √x(136 N/m / (2.1 kg + 0.013 kg))
v = ≈ 3.433 m/s