A skier is pulled by a tow rope up a frictionless ski slope that makes an angle of 12 degrees with the horizontal. The rope moves parallel to the slope with a constant speed of 1.0 m/s. The force of the rope does 880 J of work on the skier as the skier moves a distance of 7.0 m up the incline.
(a) If the rope moved with a constant speed of 2.0 m/s, how much work would the force of the rope do on the skier as the skier moved a distance of 8.0 m up the incline?
At what rate is the force of the rope doing work on the skier when the rope moves with a speed of:
(b) 1.0 m/s
(c) 2.0 m/s
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To solve this problem, we need to consider the work-energy theorem and the concept of power.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the rope on the skier is equal to the change in the skier's kinetic energy.
The change in kinetic energy (ΔKE) can be calculated using the formula:
ΔKE = KE_final - KE_initial
Since the skier starts from rest and comes to rest at the end of the ride, the final kinetic energy is zero. Therefore, the work done by the rope is equal to the initial kinetic energy of the skier:
Work = KE_initial
(a) The work done by the rope with a speed of 1.0 m/s is given as 880 J. We can calculate the initial kinetic energy using the formula:
KE_initial = Work
KE_initial = 880 J
Now, to find the work done by the rope with a speed of 2.0 m/s and a distance of 8.0 m, we need to determine the new value of the initial kinetic energy using the same formula:
KE_initial = Work
KE_initial = Work / distance * speed
KE_initial = 880 J / 7.0 m * 2.0 m/s
KE_initial = 880 J / 14.0 m
KE_initial = 62.86 J
To find the work done by the rope in this case, we can again use the work-energy theorem:
Work = KE_initial
Work = 62.86 J
Therefore, the work done by the rope with a speed of 2.0 m/s and a distance of 8.0 m is 62.86 J.
(b) To calculate the rate at which the force of the rope does work on the skier when the rope moves with a speed of 1.0 m/s, we need to calculate the power. Power is defined as the work done per unit time.
Power = Work / time
Power = Work / distance / speed
Power = Work / (7.0 m / 1.0 m/s)
Power = Work / 7.0 s
Using the given value of work (880 J), we can calculate the power:
Power = 880 J / 7.0 s
Power = 125.71 W
Therefore, the rate at which the force of the rope does work on the skier when the rope moves with a speed of 1.0 m/s is 125.71 Watts.
(c) To calculate the rate at which the force of the rope does work on the skier when the rope moves with a speed of 2.0 m/s, we use the same formula for power:
Power = Work / distance / speed
Power = Work / (8.0 m / 2.0 m/s)
Power = Work / 4.0 s
Using the value of work calculated in part (a) (62.86 J), we can calculate the power:
Power = 62.86 J / 4.0 s
Power = 15.72 W
Therefore, the rate at which the force of the rope does work on the skier when the rope moves with a speed of 2.0 m/s is 15.72 Watts.