A motorcyclist is trying to leap across a canyon by driving horizontally off a cliff (height= 60 m) at a speed of 25 m/s. The cycle strikes a cliff on the other side (height= 30 m) at a speed of 15 m/s. Find the work done by air resistance.
To find the work done by air resistance, we first need to calculate the total mechanical energy of the motorcyclist at both points: when the motorcyclist is about to leave the cliff on the first side and when the motorcyclist hits the cliff on the second side.
1. Initial point (before leaving the cliff):
At this point, the motorcyclist has gravitational potential energy due to their height above the ground, and kinetic energy due to their horizontal velocity.
The gravitational potential energy (PE) at this point is given by:
PE = m * g * h
where:
m = mass of the motorcyclist
g = acceleration due to gravity (approximately 9.8 m/s²)
h = height of the cliff (60 m)
The kinetic energy (KE) at this point is given by:
KE = (1/2) * m * v²
where:
v = velocity of the motorcyclist (25 m/s)
2. Final point (when hitting the cliff on the other side):
At this point, the motorcyclist has gravitational potential energy due to their reduced height above the ground, and kinetic energy due to their horizontal velocity (which has decreased).
The gravitational potential energy (PE) at this point is given by:
PE = m * g * h
where:
h = height of the cliff on the other side (30 m)
The kinetic energy (KE) at this point is given by:
KE = (1/2) * m * v²
where:
v = velocity of the motorcyclist (15 m/s)
Now, the work done by air resistance (W) can be calculated by subtracting the final mechanical energy (KE + PE) from the initial mechanical energy (KE + PE):
W = (KE + PE) - (KE + PE)
W = (KE₂ + PE₂) - (KE₁ + PE₁)
W = [(1/2) * m * v₂² + m * g * h₂] - [(1/2) * m * v₁² + m * g * h₁]
Substituting the given values, we can solve for W.
To find the work done by air resistance, we need to determine the change in mechanical energy of the motorcyclist as they cross the canyon.
The mechanical energy of an object can be calculated using the formula:
E = KE + PE
Where:
E = mechanical energy
KE = kinetic energy
PE = potential energy
The initial mechanical energy (E_initial) is the sum of the kinetic and potential energy before the jump, and the final mechanical energy (E_final) is the sum of the kinetic and potential energy after the jump.
E_initial = KE_initial + PE_initial
E_final = KE_final + PE_final
The change in mechanical energy (ΔE) is given by ΔE = E_final - E_initial.
First, let's calculate the initial mechanical energy:
KE_initial = (1/2) * mass * (velocity_initial)^2
PE_initial = mass * gravity * height_initial
Given:
velocity_initial = 25 m/s
height_initial = 60 m
Using the given values and assuming the mass of the motorcyclist is 1 kg for simplicity, we can calculate the initial mechanical energy:
KE_initial = (1/2) * 1 kg * (25 m/s)^2 = 312.5 J
PE_initial = 1 kg * 9.8 m/s^2 * 60 m = 588 J
E_initial = KE_initial + PE_initial = 312.5 J + 588 J = 900.5 J
Next, we need to calculate the final mechanical energy:
KE_final = (1/2) * mass * (velocity_final)^2
PE_final = mass * gravity * height_final
Given:
velocity_final = 15 m/s
height_final = 30 m
Using the given values and assuming the mass of the motorcyclist is 1 kg for simplicity, we can calculate the final mechanical energy:
KE_final = (1/2) * 1 kg * (15 m/s)^2 = 112.5 J
PE_final = 1 kg * 9.8 m/s^2 * 30 m = 294 J
E_final = KE_final + PE_final = 112.5 J + 294 J = 406.5 J
Now, we can calculate the change in mechanical energy:
ΔE = E_final - E_initial = 406.5 J - 900.5 J = -494 J
The negative sign indicates a decrease in mechanical energy, which means that work was done by external forces.
Since we are interested in the work done by air resistance, we can assume that no other external forces are acting on the motorcyclist. Therefore, the work done by air resistance is equal to the change in mechanical energy, which is -494 J.