The speed of the skier at the end of the run is approximately 176.52 m/s. The average force of resistance acting on the skier is approximately 0.952 N. Both skiers, regardless of their mass, would be traveling at the same speed at the end of the run.
(i) To find the speed of the skier at the end of the run, we can use the principle of conservation of energy.
The total mechanical energy of the skier at the start of the run is the sum of their initial kinetic energy (which is zero) and potential energy due to their altitude.
The total mechanical energy at the end of the run is the sum of the final kinetic energy and potential energy due to their new altitude.
Since resistance forces are ignored, the total mechanical energy is conserved. We can set up the equation:
Initial potential energy = Final kinetic energy + Final potential energy
m * g * h1 = 0.5 * m * v^2 + m * g * h2
where m is the mass of the skier, g is the acceleration due to gravity, h1 and h2 are the altitudes at the start and end of the run, and v is the speed of the skier at the end of the run. Substituting the given values, we get:
70 * 9.8 * 2471 = 0.5 * 70 * v^2 + 70 * 9.8 * 1863
1674178 = 0.5 * 70 * v^2 + 136734
0.5 * v^2 = (1674178 - 136734) / 70
v^2 = ((1674178 - 136734) / 70) * 2
Calculating this, we find that v^2 = 31128.74.
Taking the square root of this value, we get:
v ≈ 176.52 m/s
Therefore, the speed of the skier at the end of the run is approximately 176.52 m/s.
(ii) To find the average force of resistance acting on the skier, we can use Newton's second law of motion which states that force is equal to mass times acceleration.
In this case, the acceleration is the change in velocity divided by the time taken. We can set up the equation:
F = m * (v - u) / t
where F is the force of resistance, m is the mass of the skier, v is the final velocity, u is the initial velocity, and t is the time taken.
Substituting the given values, we get:
F = 70 * (42 - 0) / (3.1 * 1000)
F = 70 * 42 / 3100
F ≈ 0.952 N
Therefore, the average force of resistance acting on the skier is approximately 0.952 N.
(iii) Since resistance forces are ignored, the speed of the skier at the end of the run depends only on their initial potential energy and the change in altitude.
The mass of the skier does not affect their speed in this scenario.
Therefore, both skiers, regardless of their mass, would be traveling at the same speed at the end of the run.
The probable question may be:
A ski-run starts at altitude 2471 m and ends at 1863 m.
(i)If all resistance forces could be ignored, what would the speed of a skier be at the end of the run?
A particular skier of mass 70 kg actually attains a speed of 42 ms1. The length of the run is 3.1 km.
(ii) Find the average force of resistance acting on the skier.
Two skiers are equally skilful.
(iii) Which would you expect to be travelling faster by the end of the run, the heavier or the lighter?