A car rolls down a 13.0° slope for 50.0 m, accelerating at 2.90 m/s2. Leaving the cliff at 13.0° below the horizontal, with the velocity it acquired from rolling downhill, the car falls 26.0 m to the rocks below, bursting into flame upon impact.

What horizontal distance from the bottom of the cliff did the car hit the ground?

To find the horizontal distance from the bottom of the cliff where the car hits the ground, we need to break down the motion of the car into two components: horizontal and vertical.

Given:
- The car rolls down a 13.0° slope for 50.0 m, accelerating at 2.90 m/s^2.
- The car leaves the cliff at 13.0° below the horizontal with the velocity it acquired from rolling downhill.
- The car falls 26.0 m to the rocks below.

Let's start by finding the time it takes for the car to fall 26.0 m vertically. We can use the equation of motion for vertical motion:
d = v_i * t + (1/2) * a * t^2

where:
- d is the vertical distance (26.0 m),
- v_i is the initial vertical velocity (0 m/s since the car only has horizontal velocity),
- a is the vertical acceleration (-9.8 m/s^2 due to gravity),
- t is the time.

Rearranging the equation, we get:
0 = 0 * t + (1/2) * (-9.8) * t^2
0 = -4.9t^2

Solving for t, we find that t = 0 or t = √2. This means that the car takes √2 seconds to fall 26.0 m vertically.

Next, we can find the horizontal distance traveled by the car in √2 seconds. Since the car maintains its horizontal velocity when it leaves the cliff, we can use the equation for horizontal motion:
d_horizontal = v_horizontal * t

where:
- d_horizontal is the horizontal distance we are trying to find,
- v_horizontal is the horizontal velocity (which is the velocity acquired from rolling downhill),
- t is the time (√2 seconds).

We can calculate v_horizontal using the initial acceleration and distance traveled on the slope. We use the equation of motion for constant acceleration:
v_f^2 = v_i^2 + 2 * a * d

where:
- v_f is the final velocity (which is v_horizontal),
- v_i is the initial velocity (0 m/s),
- a is the acceleration (2.90 m/s^2),
- d is the distance on the slope (50.0 m).

Rearranging the equation, we get:
v_horizontal = √(0^2 + 2 * 2.90 * 50.0)

Simplifying, we find v_horizontal = 34.414 m/s.

Now we can substitute the values into the equation for horizontal distance:
d_horizontal = 34.414 * √2

Calculating this, we find:
d_horizontal ≈ 48.64 m

Therefore, the horizontal distance from the bottom of the cliff where the car hits the ground is approximately 48.64 meters.