A 42.5 g marble is fired vertically upward using a spring gun. The spring must be compressed 7.0 cm if the marble is to reach a target 27.0 m above the gun. What is the change in gravitational energy of the marble during its ascent?
Potential Energy = Kinetic Energy
m*g*h = m/2*v^2
0.0425*9.8*27 = 0.02125*v^2
Solve for v^2.
Kinetic Energy (marble) = Potential Energy (spring)
m/2*v^2 = k/2*y^2
0.02125*v^2 = k/2*0.0049^2
Solve for k.
Change in gravitational energy is equal to m*g*h.
To find the change in gravitational energy of the marble during its ascent, we need to determine the initial gravitational potential energy and the final gravitational potential energy.
The initial gravitational potential energy (U_initial) can be calculated using the formula:
U_initial = m * g * h_initial
where:
m is the mass of the marble (42.5 g = 0.0425 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h_initial is the initial height (0 m, as the marble is initially at the ground level)
U_initial = 0.0425 kg * 9.8 m/s^2 * 0 m
U_initial = 0 J
The final gravitational potential energy (U_final) can be calculated using the formula:
U_final = m * g * h_final
where:
m is the mass of the marble (0.0425 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h_final is the final height (27 m)
U_final = 0.0425 kg * 9.8 m/s^2 * 27 m
U_final = 11.6745 J
The change in gravitational energy (ΔU) is the difference between the final and initial gravitational potential energy:
ΔU = U_final - U_initial
ΔU = 11.6745 J - 0 J
ΔU = 11.6745 J
Therefore, the change in gravitational energy of the marble during its ascent is 11.6745 Joules.