Final answer:
To determine the distance Mike lands from the base of the cliff, projectile motion equations are used. The horizontal velocity is constant, and the time to fall is calculated using the vertical displacement and acceleration due to gravity. The approximate distance is calculated by multiplying the horizontal velocity by the time taken to fall, resulting in about 24.91 meters.
Step-by-step explanation:
To calculate the distance Mike lands from the base of the cliff, we will use the equations of motion for projectile motion. Since the question states that Mike runs horizontally off the cliff, his initial vertical velocity is 0 m/s.
The horizontal velocity remains constant at 8.5 m/s because there's no horizontal acceleration. The time it takes for Mike to hit the water will be determined solely by the vertical distance fallen and the acceleration due to gravity, which is 9.81 m/s2.
We first calculate the time (t) it takes to fall 42 meters using the equation for vertical motion:
s = ut + 0.5at2,
where s is the vertical displacement (42 m), u is the initial vertical velocity (0 m/s), and a is the acceleration due to gravity. Plugging in the values:
42 = 0·t + 0.5x9.81xt2
We find that t is approximately 2.93 seconds.
Now we can calculate the horizontal distance using the horizontal velocity and the time calculated:
distance = velocity x time
distance = 8.5 m/s x 2.93 s
This gives us a distance of approximately 24.91 meters from the base of the cliff.