A ski trail makes a vertical descent of 77 m. A novice skier, unable to control his speed, skis down this trail and is lucky enough not to hit any trees.
If the skier is moving at 17 m/s at the bottom of the trail, calculate the total work done by friction and air resistance during the run. The skier's mass is 74 kg.
KJ?
I don't undertand how to do this am confused :/ please help!
To calculate the total work done by friction and air resistance during the ski run, we need to understand the concept of work and the equations involved.
Work (W) is defined as the product of force and displacement, often represented by the equation: W = F * d * cos(theta), where F represents the force applied, d represents the displacement, and theta represents the angle between the force and displacement vectors.
In this scenario, the force of friction and air resistance act in the opposite direction to the skier's motion, so we will take the negative sign for the work done.
To find the work done by friction and air resistance, we need to determine the net force acting on the skier. This can be calculated using Newton's second law of motion: F = m * a, where F is the net force, m is the mass of the skier, and a is the acceleration.
Here, we are given the initial speed (vi) of the skier (17 m/s) and the final speed (vf) at the bottom of the hill (0 m/s). Since the skier comes to a stop due to the opposing forces, the change in velocity (Δv) is calculated as vf - vi = (0 - 17) m/s = -17 m/s.
We also know that the change in position or displacement (Δd) is equal to the vertical descent or height of the trail (77 m).
Next, we need to find the acceleration (a) using the second law of motion. Rearranging the equation F = m * a, we get a = F / m.
Since the net force is the force due to friction and air resistance, we can write the equation as:
F = m * a = -f_net
To find the net force (f_net), we can use Newton's second law and the equation v^2 = vi^2 + 2 * a * d. Rearranging the equation, we can solve for a:
a = (vf^2 - vi^2) / (2 * d).
Substituting in the known values, we have:
a = (-17^2 - 0^2) / (2 * 77) = -289 / 154 = -1.8766 m/s^2 (rounded to four decimal places).
Now, we can find the net force:
f_net = m * a = 74 kg * (-1.8766 m/s^2) = -138.8424 N (rounded to four decimal places).
Finally, we can calculate the work done by friction and air resistance using the negative value of the net force and the known displacement:
W = -f_net * d * cos(180°).
Since the angle between the force and displacement vectors is 180° (opposite directions), cos(180°) = -1.
Plugging in the values, we get:
W = -(-138.8424 N) * 77 m * (-1) = 10707.8928 J (rounded to four decimal places).
Converting Joules to Kilojoules, we divide by 1000:
W = 10707.8928 J / 1000 = 10.7079 KJ (rounded to four decimal places).
Therefore, the total work done by friction and air resistance during the ski run is approximately 10.7079 KJ.