how long is the skier in the air and how high is the end of the cliff

Question

The famous skier, Walajones, leaves the horizontal end of a cliff with a velocity of 40 m/s and lands 85 meters from the base of the ramp. 

a) How long is the skier in the air?

b) How high is the end of the cliff?

Answer 1

This is a constant-acceleration 2D kinematics problem.

You can divide this into two sets of kinematics variables: one corresponding to Walajones's vertical movement and the other for her horizontal movement.

First we list out all the kinematics variables (both sets). Starting with the x set, we first solve for how long the skater was in the air (Δt).

x-direction (horizontal):

Δx = 85 m

v_x0 = 40 m/s [initial horizontal velocity: given]

v_x = [we don't know the final x-velocity, and we hopefully don't need to know it.]

a_x = 0 [the only force acting on Walajones while in the air is gravity; hence there is no force in the horizontal direction, and hence no acceleration in the x-direction. Air resistance is negligible.]

Δt = ?

To find Δt (henceforth, just "t"), we'll use this kinematics equation, because it doesn't use our other unknown variable, v_x:

Δx = v_x0t + 0.5a_xt².

Substituting in our known variables:

85 = 40t

And finally, solving for t:

t = 85/40 = 2.125 s

It takes Walajones 2.125 seconds to land at 85 m beyond the cliff.

y-direction (vertical)

Δy = ?

v_y0 = 0 m/s [because at first Walajones was only moving horizontally along the cliff, not vertically.]

v_yf =

a_y = -9.8 m/s² [constant acceleration of gravity]

Δt = ? = 2.125 s

Note that Δt here is the same Δt as in the x set of variables, because the two sets of variables correspond to the same events occurring in the same span of time.

Now, our task is to solve for the height of the cliff, Δy, which is the vertical distance Walajones travels in time t. We again use the equation that does not involve v_yf because we don't want to do extra calculations on another unknown variable:

Δy = v_y0t + 0.5a_yt².

Substituting in our known variables:

Δy = -4.9(2.125)² ≈ -22.127 m

Δy is negative because we have solved for the vertical distance that Walajones has travelled; and she has travelled downwards, so it is a negative distance. The height of the cliff is simply positive 22.127m.

Let me know if you find any errors in this solution!