Question

In a downhill ski race, surprisingly, little advantage is gained by getting a running start. (This is because the initial kinetic energy is small compared with the gain in gravitational potential energy even on small hills.) To demonstrate this, find the final speed in m/s and the time taken in seconds for a skier who skies 65.0 m along a 25° slope neglecting friction for the following two cases. (Enter the final speeds to at least one decimal place.)

(a) starting from rest final speed m/s time taken ____s

(b) starting with an initial speed of 3.20 m/s final speed m/s time taken ____s

(c) Does the answer surprise you? Discuss why it is still advantageous to get a running start in very competitive events.

Answer 1

Height of the hill be H
H/65 = sin(25)
H = 65 x sin(25) = 27.47 m

a)
Total energy at the top = mgH = 269.208 m.
Total energy at the bottom = 1/2 mv²
Equating both energies,
269.208 m = 1/2 mv²
v = SQRT[538.42]
= 23.2 m/s

Using the formula, v = u + at,
u = 0
a = g sin(25) = 4.142 m/s².
t = v/a
= 23.2 / 4.142
= 5.60 s

b)
Total energy at the top = 1/2 m (3.2)² + m x 9.8 x 27.47
= 274.33 m
Total energy at the bottom = 1/2 mv²
Equating both the energies,
274.33 m = 1/2 mv²
v = SQRT[548.66]
= 23.42 m/s

Here initial velocity, u = 3.2 m/s
a = g sin(25) = 4.142 m/s²
t = (v - u)/a
= (23.42 - 3.2) / 4.142
= 4.88 s

c)
The increase of velocity in the second case is small. Time taken to come down decreased by 0.7 s, which can decide the results in competitive events.