You accidentally throw your car keys horizontally at 10.0 m/s from a cliff 78 m high. How far from the base of the cliff should you look for the keys?

how much time does it take to fall 78m?

hf=1/2 g t^2
-78=1/2 9.8 t^2 solve for t.

how far horizontally? distance= 10m/s * time

The time required fir the keys to to fall H = 78 m is

T = sqrt(2H/g) = 3.99 seconds.

Multiply that by the initial velocity, which remains the constant horizontal velocity component during the fall to the ground below.

To determine how far from the base of the cliff you should look for the keys, you need to calculate the horizontal distance traveled by the keys. Let's break down the problem into steps:

Step 1: Analyze the information given:
- Initial velocity in the horizontal direction (vx) = 10.0 m/s
- Vertical displacement (y) = -78 m (negative because the keys are thrown downward)
- Acceleration due to gravity (g) = 9.8 m/s^2 (acting vertically downward)

Step 2: Calculate the time of flight:
Since the vertical and horizontal motions are independent, we can first calculate the time it takes for the keys to fall from the cliff before they reach the ground.

Use the equation for vertical displacement:
y = (1/2) * g * t^2

Rearrange the equation to solve for time (t):
t^2 = (2 * y) / g
t = sqrt((2 * y) / g)

Substitute the values:
t = sqrt((2 * -78 m) / 9.8 m/s^2)
t = sqrt(-15.9184 s^2)
(Note: We have a negative sign because the vertical displacement is downward.)

Since time cannot be negative in this context, we discard the negative sign, and we find that the keys take approximately 3.99 seconds to fall from the cliff.

Step 3: Calculate the horizontal distance:
Use the equation for horizontal distance:
x = vx * t

Substitute the values:
x = (10.0 m/s) * (3.99 s)
x ≈ 39.9 m

Therefore, you should look approximately 39.9 meters from the base of the cliff for your car keys.