In an action movie, the heroine is supposed to jump with a motorcycle from the roof of one skyscraper to another one. The roof of the second skyscraper is h=9 meters lower, and the gap between the buildings is d=15 meters wide. Both roofs are horizontal and flat. Neglecting air resistance, what minimum initial speed would she need to make the jump?

Question

In an action movie, the heroine is supposed to jump with a motorcycle from the roof of one skyscraper to another one. The roof of the second skyscraper is h=9 meters lower, and the gap between the buildings is d=15 meters wide. Both roofs are horizontal and flat. Neglecting air resistance, what minimum initial speed would she need to make the jump?

Answer 1

To calculate the minimum initial speed the heroine would need to make the jump between the skyscrapers, we can use the principles of projectile motion.

First, let's break down the given information:
- The height difference between the two skyscrapers is h = 9 meters.
- The horizontal distance, or gap, between the buildings is d = 15 meters.

Now, let's identify the key components of projectile motion for the motorcycle jump:
1. Vertical motion: The motorcycle will undergo free fall due to gravity while traveling vertically.
2. Horizontal motion: The motorcycle will maintain a constant speed horizontally.

To calculate the minimum initial speed needed, we can use the equations of motion:

1. Vertical motion equation:
h = ut - (1/2)gt^2
where h is the height difference, u is the initial vertical velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time taken to reach the other skyscraper.

Since we want to calculate the minimum initial speed, we can assume that the motorcycle lands on the edge of the other roof at the same time it falls vertically. Therefore, the time taken for vertical motion is the same as the time taken for horizontal motion.

2. Horizontal motion equation:
d = vt
where d is the horizontal distance, v is the initial horizontal velocity, and t is the time taken for both vertical and horizontal motion.

Since we want to find the minimum initial speed, the motorcycle should barely make it to the other roof. In this scenario, the vertical displacement is equal to the height difference between the buildings.

So, we can rewrite the vertical motion equation as:
h = (1/2)gt^2

To solve for t, we rearrange the equation:
t = sqrt(2h / g)

Now, substitute this value of t into the horizontal motion equation:
d = v * sqrt(2h / g)

Rearrange this equation to solve for v:
v = d / sqrt(2h / g)

Finally, substituting the given values:
v = 15 / sqrt(2 * 9 / 9.8)

Calculating this, we get:
v ≈ 15.652 m/s

Therefore, the minimum initial speed the heroine would need to make the jump is approximately 15.652 m/s.