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 70.0 m along a 25° 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 3.00 m/s final speed_________m/s

time taken_________s

Explanation / Answer

(a)
Using Energy Conservation,
Initial Kinetic Energy + Initial Potential Energy = Final Kinetic Energy + Final Potential Energy
0 + m*g*h = 1/2*mv^2 + 0
9.8 * 75*sin(25) = 1/2* v^2
v = 24.9 m/s

Calculating acceleration,
v^2 = u^2 + 2*a*s
24.9^2 = o + 2*a*70.0
a = 4.43 m/s^2

v = u + a*t
t = v/a
t = 24.9/4.43
t = 5.62 s
Time taken, t = 5.62 s

(b)
Using Energy Conservation,
Initial Kinetic Energy + Initial Potential Energy = Final Kinetic Energy + Final Potential Energy
1/2*mu^2 + m*g*h = 1/2*mv^2 + 0
1/2*3.0^2 + 9.8 * 75*sin(25) = 1/2* v^2
v = 25.1 m/s


Calculating acceleration,
v^2 = u^2 + 2*a*s
24.9^2 = 3^2 + 2*a*70.0
a = 4.36 m/s^2

v = u + a*t
t = (v-u)/a
t = (25.1- 3.0)/4.36
Time taken, t = 5.07 s

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