In the flying ski jump, a skier acquires a speed of about 62 mph (this ended up being 27.7 m/s) by racing down a steep hill and then lifts off into the air off of a horizontal ramp. Beyond the ramp, the ground slopes at an angle of 45 degrees.
a) Predict the distance downslope from the ramp that the skier lands.
35.4m
To predict the distance downslope from the ramp that the skier lands, we can use the equations of linear motion. Let's break down the problem step by step.
1. Determine the initial velocity of the skier:
Given that the skier has a speed of 27.7 m/s.
2. Resolve the velocity into its horizontal and vertical components:
Since the skier lifts off from a horizontal ramp, the initial horizontal velocity remains constant. The initial vertical velocity is zero.
3. Calculate the time taken by the skier to reach the highest point of the jump:
We'll use the equation: vf = vi + at, where vf is the final vertical velocity (zero since it reaches its peak), vi is the initial vertical velocity (zero), a is the vertical acceleration (due to gravity, -9.8 m/s^2), and t is the time taken.
Rearranging the equation, we find t = (vf - vi) / a.
4. Calculate the maximum height reached by the skier:
We'll use the equation: s = vit + (1/2)at^2, where s is the vertical displacement, vi is the initial vertical velocity, t is the time taken (from step 3), and a is the vertical acceleration.
Rearranging the equation, we find s = vi*t + (1/2)at^2.
5. Calculate the total time of flight:
The total time of flight is twice the time taken to reach the highest point, as the time taken to ascend is equal to the time taken to descend.
6. Calculate the horizontal distance covered during flight:
The horizontal distance is given by the equation: d = v*t, where d is the horizontal distance, v is the horizontal velocity (27.7 m/s), and t is the total time of flight (from step 5).
7. Calculate the distance downslope from the ramp that the skier lands:
The distance downslope can be calculated using the equation: distance downslope = horizontal distance * tan(angle), where angle is the slope angle (45 degrees).
Plugging the relevant values into the equations, we can determine the distance downslope from the ramp that the skier lands.