A 70kg bicyclist, initially at rest at the top of a hill, coasts down the hill. During her ride she experiences a 40N drag. The distance and height of the hill are shown. What’s the loss J
To calculate the loss in mechanical energy, we need to find the difference in gravitational potential energy at the top and bottom of the hill.
The loss in mechanical energy can be calculated using the formula:
J = ΔPE + ΔKE + ΔW
where:
J is the loss in mechanical energy
ΔPE is the change in gravitational potential energy
ΔKE is the change in kinetic energy
ΔW is the work done against the drag force
First, let's calculate the change in gravitational potential energy:
ΔPE = m * g * Δh
where:
m is the mass of the bicyclist (70 kg)
g is the acceleration due to gravity (9.8 m/s^2)
Δh is the change in height (90 m)
ΔPE = 70 kg * 9.8 m/s^2 * 90 m = 62,460 J
Next, let's calculate the change in kinetic energy:
ΔKE = 1/2 * m * Δv^2
where:
m is the mass of the bicyclist (70 kg)
Δv is the change in velocity
Since the bicyclist starts at rest and experiences a drag force of 40 N, we can use Newton's second law to find the acceleration:
F = m * a
40 N = 70 kg * a
a = 40 N / 70 kg = 0.5714 m/s^2
Next, let's use the kinematic equation to find the change in velocity:
Δv^2 = 2 * a * Δd
where:
a is the acceleration (0.5714 m/s^2)
Δd is the change in distance (300 m)
Δv^2 = 2 * 0.5714 m/s^2 * 300 m = 343.7 m^2/s^2
ΔKE = 1/2 * 70 kg * 343.7 m^2/s^2 = 12,020.5 J
Finally, let's calculate the work done against the drag force:
ΔW = F * Δd
where:
F is the drag force (40 N)
Δd is the change in distance (300 m)
ΔW = 40 N * 300 m = 12,000 J
Now, let's plug these values into the formula for the loss in mechanical energy:
J = ΔPE + ΔKE + ΔW
J = 62,460 J + 12,020.5 J + 12,000 J
J ≈ 86,480.5 J
Therefore, the loss in mechanical energy during the ride down the hill is approximately 86,480.5 J.