A parachutist with mass 73.9 kg jumps from an airplane traveling at a speed 𝑣𝑖=112 km/hr at a height 𝐻=2570 m. He lands with a speed of 𝑣𝑓=5.21 m/s. What is the change in mechanical energy of the earth-parachutist system from just after the jump to just before landing?
start: PE=mgh = 73.9 * 9.81 * 2570
KE = 1/2 mv^2 = 1/2 * 73.9 * (112 km/hr * 1000m/km * 1hr/3600s)^2
end: PE = mgh = 0
KE = 1/2 mv^2 = 1/2 * 73.9 * 5.21^2
To determine the change in mechanical energy of the earth-parachutist system, we need to calculate the initial and final mechanical energies and then find their difference. The mechanical energy is the sum of the kinetic energy and potential energy.
1. Calculate the initial mechanical energy:
The initial kinetic energy is given by the equation K = 0.5 * m * v^2, where m is the mass and v is the speed. Convert the initial speed, 𝑣𝑖, from km/hr to m/s by dividing it by 3.6.
K_initial = 0.5 * 73.9 kg * (𝑣𝑖/3.6)^2
The initial potential energy is given by the equation U = m * g * h, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
U_initial = 73.9 kg * 9.8 m/s^2 * 2570 m
The total initial mechanical energy is the sum of the initial kinetic energy and potential energy:
E_initial = K_initial + U_initial
2. Calculate the final mechanical energy:
The final kinetic energy is given by the equation K = 0.5 * m * v^2, where m is the mass and v is the speed.
K_final = 0.5 * 73.9 kg * 𝑣𝑓^2
The final potential energy is zero because the parachutist is on the ground, and the height is considered to be zero.
U_final = 0
The total final mechanical energy is the sum of the final kinetic energy and potential energy:
E_final = K_final + U_final
3. Calculate the change in mechanical energy:
The change in mechanical energy is given by the equation ΔE = E_final - E_initial
ΔE = E_final - E_initial
Now, plug in the values calculated above to find the change in mechanical energy, ΔE.
To find the change in mechanical energy of the earth-parachutist system, we need to calculate the initial and final mechanical energies and then subtract them.
The initial mechanical energy of the system (just after the jump) can be calculated as the sum of the potential energy and the kinetic energy:
Initial mechanical energy = Potential energy + Kinetic energy
The potential energy can be calculated using the formula:
Potential energy = mass × gravity × height
where:
mass = 73.9 kg (mass of the parachutist)
gravity = 9.8 m/s^2 (acceleration due to gravity on Earth)
height = 2570 m (height of the airplane)
Potential energy = 73.9 kg × 9.8 m/s^2 × 2570 m
The kinetic energy can be calculated using the formula:
Kinetic energy = (1/2) × mass × velocity^2
where:
mass = 73.9 kg (mass of the parachutist)
velocity = 112 km/hr = (112 × 1000) m/3600 s = 31.11 m/s (initial velocity of the parachutist)
Kinetic energy = (1/2) × 73.9 kg × (31.11 m/s)^2
Now, let's calculate the initial mechanical energy:
Initial mechanical energy = Potential energy + Kinetic energy
Initial mechanical energy = (73.9 kg × 9.8 m/s^2 × 2570 m) + (1/2) × 73.9 kg × (31.11 m/s)^2
Next, we calculate the final mechanical energy of the system (just before landing). The final mechanical energy is equal to the sum of the potential energy and the kinetic energy at this point.
The potential energy is the same as before, as it depends only on the height. So we can use the same calculation:
Potential energy = 73.9 kg × 9.8 m/s^2 × 2570 m
The final kinetic energy can be calculated using the formula:
Kinetic energy = (1/2) × mass × velocity^2
where:
mass = 73.9 kg (mass of the parachutist)
velocity = 5.21 m/s (final velocity of the parachutist)
Kinetic energy = (1/2) × 73.9 kg × (5.21 m/s)^2
Now, let's calculate the final mechanical energy:
Final mechanical energy = Potential energy + Kinetic energy
Final mechanical energy = (73.9 kg × 9.8 m/s^2 × 2570 m) + (1/2) × 73.9 kg × (5.21 m/s)^2
Finally, we can find the change in mechanical energy by subtracting the final mechanical energy from the initial mechanical energy:
Change in mechanical energy = Final mechanical energy - Initial mechanical energy
Substitute the values into the equation to find the change in mechanical energy.