2D Kinematics Problem

Question

Wile E. Coyote is in an Acme Rocket Sled traveling at a velocity of 100 m/s when he misses a turn and flies off a 46-meter high cliff. How fast does Wile crash into the ground. My question is how do I do this problem if there seems to not be enough information for either the x equations or the y equations?

Answer 1

Our strategy here is to first figure out how long Wile E. Coyote is in the air before he hits the ground. If we know that time, then we should be able to figure out his final velocities in the horizontal and vertical directions, which then would give us enough information to find the final speed.

Let's look at the vertical velocity first:

The initial velocity in the vertical direction is v_0,y= 0 m/s (since he is only moving horizontally at the instant he goes off the cliff). His initial height is x_0,y= 46 m. We also know that the acceleration in the y direction is equal to a_y = -9.8 m/s² (since we assume the only force acting in the vertical direction is gravity). We're interested in the time t he hits the ground, so let's set his position at time t as x_t,y = 0.

Using kinematic equations, we know that x_f,y = (1/2)a_yt²+v_0,yt+x_0,y . Substituting everything we know gives an equation 0 = (1/2)(-9.8)t²+(0)t+46 = -4.9t²+46. Solving this quadratic equation gives t ≈ 3.06 s.

Going back to our kinematic equations, we know that v_t,y = v_0,y+a_yt. Substituting the value of t along with the acceleration and initial velocity gives v_t,y ≈ 0+ (-9.8) (3.06) ≈ -30 m/s (remember that this is approximate, since we are using rounded values for the time as well as the acceleration due to gravity - you might need to use more decimal places if you need a more accurate answer).

Now that we know the vertical component of velocity when Wile E. Coyote hits the ground, we need to find the horizontal component. This part is easy, since there are no forces acting in the horizontal direction, so the horizontal component when he hits the ground at time t is just equal to the initial horizontal velocity: v_x,t= 100 m/s.

We know now that, when Wile E. Coyote hits the ground at time t, the horizontal component is v_x,t= 100 m/s and the vertical component is v_t,y≈ -30 m/s. All that's left is to use what we know about vectors to find the magnitude of the velocity (speed is equal to the magnitude of velocity) at time t. I'll leave it to you to finish up and arrive at your final answer.