A 44 kg skier starts from rest at a height of H = 18 m above the end of the ski-jump ramp. As the skier leaves the ramp, his velocity makes an angle of 28° with the horizontal. Neglect the effects of air resistance and assume the ramp is frictionless.
(a) What is the maximum height h of his jump above the end of the ramp?
m
Speed acquired at end of ramp
Ve = sqrt(2 g H) = 18.73 m/s
That can be derived using conservation of energy.
The upward vertical component of velocity at the edge of the ski jump is
Vy = Ve*sin28 = 8.79 m/s
That is enough to cause a vertical rise (above the end of the jump) equal to
y = Vy^2/(2*g) = 3.94 m
To find the maximum height h of the skier's jump above the end of the ramp, we can use the principle of conservation of mechanical energy. The initial potential energy of the skier at height H is converted into kinetic energy as the skier leaves the ramp and then if there are no external forces doing work, the final kinetic energy is converted back into potential energy at the maximum height.
The initial potential energy is given by the formula:
Potential Energy = mass * gravity * height
where mass is the mass of the skier (44 kg), gravity is the acceleration due to gravity (9.8 m/s^2), and height is the initial height above the end of the ramp (H = 18 m).
Initial Potential Energy = (44 kg) * (9.8 m/s^2) * (18 m)
Next, we need to find the final kinetic energy of the skier at the maximum height. The skier leaves the ramp with a velocity that makes an angle of 28° with the horizontal. We can decompose this velocity into its vertical and horizontal components.
Vertical velocity = velocity * sin(angle)
Horizontal velocity = velocity * cos(angle)
Since the skier started from rest, the initial velocity is zero. At the maximum height, the vertical velocity will also be zero. So, we can set the vertical component of the velocity equal to zero:
Vertical velocity = 0 = velocity * sin(angle)
Solving this equation for velocity, we get:
velocity = 0 / sin(angle) = 0
Since the velocity is zero at the maximum height, the final kinetic energy is also zero.
The final potential energy at the maximum height is given by the formula:
Potential Energy = mass * gravity * height
where mass is the mass of the skier (44 kg), gravity is the acceleration due to gravity (9.8 m/s^2), and height is the maximum height above the end of the ramp (h).
Final Potential Energy = (44 kg) * (9.8 m/s^2) * h
According to the principle of conservation of mechanical energy, the initial potential energy must equal the final potential energy:
(44 kg) * (9.8 m/s^2) * 18 m = (44 kg) * (9.8 m/s^2) * h
Now, we can solve this equation for h:
h = (18 m)
Therefore, the maximum height h of the skier's jump above the end of the ramp is 18 m.