A 79.5 kg snowboarder heads down a 28.0° hill that has a height of 62.8 m. If the hill is assumed to be frictionless and there is horizontal wind with a force of 98 N acting against the snowboarder, find the speed of the snowboarder as they reach the bottom of the hill using work and energy.
So here is what I did
W = Fdcos(theta)
= 98 * 62.8* cos(28)= 5434.01
Then
v = sqrt (2W/m) = 11.7 m/s
I'm not sure what I did is correct... Please help...
Well, I would have used energy
loss of potential energy = gain in kinetic energy + work done by wind
loss of potential energy = m g h
= 79.5 (9.81) (62.8) Joules
= 48,977 Joules
work done by wind
component of horizontal wind opposite direction of motion = 98 cos 28 Newton
distance traveled = 62.8/sin 28 meter
so work done by wind = 98*62.8*cot 28
= 11,574 Joule
so
(1/2) m v^2 = 48,977 - 11,574
Your approach to solving this problem is close, but there are a few errors in your calculations.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done on the snowboarder is equal to the change in their gravitational potential energy as they descend the hill.
To find the work done by the wind force, we need to calculate the component of the force that is parallel to the direction of motion. In this case, the force is acting against the motion, so its component would be negative. Therefore, the work done by the wind force is calculated as:
Work_wind = F_wind * d * cos(180°) = -98 * 62.8 * cos(180°) = -6216.8 J
Next, we can calculate the change in gravitational potential energy as the snowboarder goes from the top to the bottom of the hill. This can be calculated using:
ΔPE = m * g * h * cos(θ) = 79.5 * 9.8 * 62.8 * cos(28°) = 33839.66 J
Since the hill is assumed to be frictionless, there are no other forms of work involved.
According to the work-energy principle, the work done on the snowboarder is equal to the change in their total mechanical energy. In this case, that would be:
Work_total = ΔKE + ΔPE = ΔKE - mgh = 33839.66 J
To find the speed of the snowboarder at the bottom of the hill, we can equate the work done to the final kinetic energy:
Work_total = 1/2 * m * v^2
33839.66 J = 1/2 * 79.5 kg * v^2
Simplifying and solving for v:
v^2 = (33839.66 J * 2) / 79.5 kg = 852.09 m^2/s^2
v = sqrt(852.09 m^2/s^2) = 29.19 m/s
Therefore, the speed of the snowboarder as they reach the bottom of the hill is approximately 29.19 m/s.