A criminal is escaping across a rooftop and runs off the roof horizontally at a speed of 4.6 m/s, hoping to land on the roof of an adjacent building. Air resistance is negligible. The horizontal distance between the two buildings is D, and the roof of the adjacent building is 2.0 m below the jumping-off point. Find the maximum value for D.
See previous post.
To find the maximum value for D, we need to determine the horizontal distance the criminal can jump before landing on the roof of the adjacent building.
Let's break down the problem into two separate motions: the criminal's horizontal motion and vertical motion.
1. Horizontal Motion:
Since there is no air resistance in this problem, the horizontal motion is characterized by a constant velocity. The criminal runs horizontally at a speed of 4.6 m/s.
2. Vertical Motion:
The criminal jumps from the roof with an initial vertical speed of 0 m/s. The only force acting on the criminal in the vertical direction is gravity, causing the criminal to accelerate downward with a magnitude of 9.8 m/s².
Now, we can find the time it takes for the criminal to hit the roof of the adjacent building.
First, we'll consider the vertical motion. We can use the following equation to calculate the time it takes to reach the roof of the adjacent building:
Δy = v₀t + (1/2)at²
Where Δy is the vertical distance traveled, v₀ is the initial vertical velocity, t is time, and a is acceleration in the vertical direction.
In this case, v₀ is 0 m/s, Δy is -2 m (negative because the criminal is jumping downward), and a is -9.8 m/s². Plugging in these values into the equation, we get:
-2 = 0 * t + (1/2) * (-9.8) * t²
-2 = -4.9t²
We can now solve this quadratic equation for t.
Rearranging the equation, we have:
4.9t² = 2
Dividing both sides by 4.9, we get:
t² = 2 / 4.9
t² ≈ 0.408
Taking the square root of both sides, we find:
t ≈ ± 0.639 seconds
Since time can't be negative in this context, we take the positive value:
t ≈ 0.639 seconds
Now that we have the time it takes for the criminal to reach the roof of the adjacent building, we can find the horizontal distance traveled (D) using the following equation:
D = v * t
Where D is the horizontal distance, v is the horizontal velocity, and t is the time.
Plugging in the values, we have:
D = 4.6 m/s * 0.639 s
D ≈ 2.94 meters
Therefore, the maximum value for D, the horizontal distance the criminal can jump, is approximately 2.94 meters.