A freestyle skier hits the take off ramp with an initial velocity of 17.5 m/s with a purely vertical take off (90 degrees). What is the peak height that the skier will reach?
Rise time = (17.5 m/s)/g = 1.79 s
Vertical distance travelled
= (Vaverage)*t
= (1/2)(17.5)(1.79)
= 15.6 meters
To find the peak height that the skier will reach, we can use the principles of projectile motion. We know that the initial velocity in the vertical direction is 17.5 m/s and the takeoff angle is 90 degrees (purely vertical).
In projectile motion, we can split the motion into horizontal and vertical components. Since the takeoff angle is purely vertical, there is no horizontal velocity component, and the entire initial velocity of 17.5 m/s is in the vertical direction.
The key equation we can use to find the peak height is the vertical motion equation:
h = (v^2 * sin^2θ)/(2g)
where,
h = peak height
v = initial velocity (17.5 m/s)
θ = launch angle (90 degrees)
g = acceleration due to gravity (9.8 m/s^2, assuming Earth's gravity)
Now let's substitute the given values into the equation:
h = (17.5^2 * sin^2(90))/(2 * 9.8)
To evaluate sin^2(90), we need to convert the angle from degrees to radians. Since sin(90 degrees) = 1, sin^2(90) will also be 1.
h = (17.5^2 * 1)/(2 * 9.8)
h = (306.25)/(19.6)
h ≈ 15.63 meters
Therefore, the peak height that the skier will reach is approximately 15.63 meters.