A spring-loaded toy gun is used to shoot a ball of mass m straight up in the air. The spring has spring constant k. If the spring is compressed a distance x_0 from its equilibrium position ( y = 0 ) and then released, the ball reaches a maximum height h_max (measured from the equilibrium position of the spring). There is no air resistance, and the ball never touches the inside of the gun. Assume x_0 is less then h_max

Initially, the spring was compressed a distance x_0; its total initial energy was E_i = (1/2)kx_0^2 (neglecting the potential energy from the small change in height, x_0).

Find the total mechanical energy of the ball when it is a height h above the equilibrium position of the spring. Assume that h < h_max , so that the ball has some velocity v. Define the gravitational potential energy to be zero at the equilibrium height of the spring.
Express the total mechanical energy in terms of h, v, g, and the ball's mass m.

Can you help me

You simply add potential and kinetic energies since it has not reached its full potential height.

E=(1/2)*m*v^2+m*g*h

To find the total mechanical energy of the ball at a height h above the equilibrium position of the spring, we need to consider the potential energy and kinetic energy of the ball.

The potential energy of the ball at height h can be calculated as the gravitational potential energy relative to the equilibrium height of the spring. Since the gravitational potential energy is zero at the equilibrium height, we can define the potential energy at height h as:

Potential energy = mgh

where m is the mass of the ball, g is the acceleration due to gravity, and h is the height above the equilibrium position of the spring.

The kinetic energy of the ball at height h can be calculated using the equation:

Kinetic energy = (1/2)mv^2

where m is the mass of the ball and v is the velocity of the ball at height h.

Therefore, the total mechanical energy of the ball at height h can be obtained by adding the potential energy and kinetic energy:

Total mechanical energy = Potential energy + Kinetic energy
= mgh + (1/2)mv^2

So, the total mechanical energy in terms of h, v, g, and the ball's mass m is:

Total mechanical energy = mgh + (1/2)mv^2

cheater