Final answer:
The question asks for the speed of a mass attached to a vertical spring after being pulled upwards a certain distance. Work-energy principle is used to solve for the speed, factoring in the work done by the rope's tension and the change in the spring's potential energy, as well as the work done against gravity.
Step-by-step explanation:
To solve for the speed of the block after it's been pulled upward over a distance of 1.30 m, we apply the work-energy principle. The work done by the tension force goes into the elastic potential energy stored in the spring and the kinetic energy of the block.
The initial elastic potential energy in the spring is given by U = (1/2)kx^2, where k is the spring constant and x is the compression distance. However, we're not given x directly, so we'll need to calculate it using the force of the tension applied, which is 480 N and is equal to kx (because the spring is already in equilibrium due to this force).
Once x is found, we can use U to find the initial potential energy. As the block is pulled upwards, the spring does work W = U - (1/2)k(x - 1.30 m)^2, where (x - 1.30 m) is the new compression of the spring after being pulled up. Also, we must consider the work done against gravity, which is Wg = -mg(1.30 m). The net work done on the block is then Wnet = W - Wg and this net work is equal to the kinetic energy of the block, so we can solve for the block's final speed using K = (1/2)mv^2.