In a downhill ski race surprisingly little advantage is gained by getting a runn

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 and the time taken for a skier who skies 75.0 m along a 30° slope neglecting friction for the following two cases. (Note that this time difference can be very significant in competitive events so it is still worthwhile to get a running start.)

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

(b) starting with an initial speed of 2.50 m/s
final speed
        = m/s
time taken
         = s

Explanation / Answer

a) for the skier starting from rest

Using conseravtion of energy

mg* d * sin(theta) = 0.5 mv^2

9.8 * 75 * sin(30) = 0.5 * v^2

v = 27.11 m/s

the final speed is 27.11 m/s

for the time taken is t

v = u + a * t

27.11 = 9.8 * sin(30) * t

t = 5.53 s

the time taken is 5.53 s

b)

for the initial speed , u = 2.5 m/s

using conseravtion of energy

mg* d * sin(theta) = 0.5 mv^2 - 0.5 * m * u^2

9.8 * 75 * sin(30) = 0.5 * v^2 + 0.5 * 2.5^2

solving for v

v = 27.22 m/s

the final speed is 27.22 m/s

for the time taken

75 = 2.5 * t + 0.5 * 9.8 * sin(30) * t^2

solving for t

t = 5.05 s

the time taken is 5.05 s

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