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=8 meters lower, and the gap between the buildings is d=16 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=8 meters lower, and the gap between the buildings is d=16 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 find the minimum initial speed the heroine would need to make the jump, we can use the principles of projectile motion.

First, let's consider the horizontal motion. The only force acting on the motorcycle in the horizontal direction is its initial velocity. We know that the gap between the buildings is d=16 meters wide, so the time it takes to travel horizontally is the same for any trajectory.

Using the equation for horizontal motion, we can find the time it takes to travel across the gap:
time = distance / horizontal velocity
time = 16 meters / horizontal velocity

Now, let's consider the vertical motion. The only force acting on the motorcycle in the vertical direction is gravity. We want to find the minimum initial speed, so we need to determine the speed that will allow the motorcycle to just clear the height difference.

Using the equation for vertical motion, we can find the initial vertical velocity required to clear the height difference:
height difference = (initial vertical velocity * time) + (0.5 * gravitational acceleration * time^2)
-8 meters = (initial vertical velocity * time) + (0.5 * gravitational acceleration * time^2)

Since we're neglecting air resistance, the only acceleration acting on the motorcycle is the acceleration due to gravity. We can assume the value of gravitational acceleration as approximately 9.8 m/s^2.

Now, we have two equations: one for horizontal motion (time = 16 meters / horizontal velocity) and one for vertical motion (-8 meters = (initial vertical velocity * time) + (0.5 * gravitational acceleration * time^2)).

We can substitute the value of time from the horizontal motion equation into the vertical motion equation to eliminate the variable "time".

After substituting, we have a quadratic equation in terms of initial vertical velocity. We can solve this equation to find the minimum initial vertical velocity required.

Finally, the minimum initial speed would be the magnitude of the vector sum of the initial horizontal velocity and the initial vertical velocity.

Please note that the values provided in the question are necessary to solve these equations.