A pumpkin with a mass of 3.0 kg is launched from a catapult at an initial height of 3.5 m off the ground, with an initial speed of 16.2 m/s. What height is the pumpkin at when the speed is half its initial speed?
To determine the height of the pumpkin when it is traveling at half its initial speed, we will make use of the principle of conservation of energy.
First, let's find the potential energy (PE) of the pumpkin at its initial height:
PE = m * g * h
Where:
m = mass of the pumpkin = 3.0 kg
g = acceleration due to gravity = 9.8 m/s²
h = initial height = 3.5 m
PE = 3.0 kg * 9.8 m/s² * 3.5 m
PE = 102.9 J (Joules)
Next, let's calculate the kinetic energy (KE) of the pumpkin when its speed is half its initial speed. Since energy is conserved, we can equate the initial potential energy to the sum of the final potential energy and the final kinetic energy:
PE_initial = PE_final + KE_final
Since the final kinetic energy is half the initial kinetic energy, we can express it as:
KE_final = 0.5 * KE_initial
The initial kinetic energy is given by:
KE_initial = 0.5 * m * v_initial^2
Where:
m = mass of the pumpkin = 3.0 kg
v_initial = initial speed = 16.2 m/s
KE_initial = 0.5 * 3.0 kg * (16.2 m/s)^2
KE_initial = 391.86 J (Joules)
Substituting the values into the conservation of energy equation:
PE_initial = PE_final + KE_final
102.9 J = PE_final + 0.5 * 391.86 J
Rearranging the equation to solve for PE_final:
PE_final = 102.9 J - 0.5 * 391.86 J
PE_final = 102.9 J - 195.93 J
PE_final = -93.03 J (Negative value indicates the height is below the reference point)
The potential energy at the final height is negative because it is measured below the reference point (ground level). To find the height (h_final), we can rearrange the potential energy equation:
PE_final = m * g * h_final
Rearranging the equation to solve for h_final:
h_final = PE_final / (m * g)
h_final = -93.03 J / (3.0 kg * 9.8 m/s²)
h_final ≈ -1.0 m
The negative sign indicates that the height is below the reference point, which in this case, is the initial height (3.5 m) above the ground. Therefore, the pumpkin is at a height of approximately 2.5 meters (3.5 m - 1.0 m) when its speed is half its initial speed.