A daredevil decides to jump a canyon of width 6.5 m. To do so, he drives a motorcycle up an incline sloped at an angle of 9.45◦.

The acceleration of gravity is 9.8 m/s2 .
What minimum speed must he have in or- der to clear the canyon?
Answer in units of m/s

Assuming both sides of the canyon are at the same elevation, then use the range equation:

range = vi²sin(2θ)/g
6.5=vi²sin(2*9.45°)/9.8
vi=sqrt(6.5*9.8/sin(18.9°)
=14.0 m/s

To find the minimum speed required to clear the canyon, we need to analyze the motion of the daredevil when he takes off from the incline.

Let's break down the problem into two components: horizontal and vertical.

1. Horizontal Component:
The width of the canyon is 6.5 m, and the daredevil needs to clear it. Since there is no horizontal force acting on the motorcycle after it leaves the incline, the horizontal speed remains constant. Therefore, the horizontal component of the daredevil's velocity (Vx) will be the same before and after the jump.

2. Vertical Component:
The daredevil will be subject to two forces in the vertical direction: gravitational force and the component of the motorcycle's weight along the incline. Let's calculate the vertical speed (Vy) at the moment the motorcycle leaves the incline.

The gravitational force acting on the motorcycle can be calculated using the equation:
Fg = m * g,
where Fg is the force due to gravity, m is the mass of the daredevil and the motorcycle, and g is the acceleration due to gravity (9.8 m/s^2).

The vertical component of the force due to gravity can be found by multiplying Fg by sin(θ), where θ is the angle of the incline (9.45°):
Fg_vertical = Fg * sin(θ)

Since there is no vertical acceleration once the daredevil leaves the incline, the vertical component of the daredevil's initial velocity (Vy) should be equal to the component of the motorcycle's weight along the incline.

Now, we can calculate Vy:
Vy = Fg_vertical = m * g * sin(θ)

Finally, to clear the canyon, the daredevil must have enough vertical speed to reach a certain height (h), given by the equation:
h = (Vy^2) / (2 * g)

We know that h is equal to the width of the canyon, which is 6.5 m. Therefore, we can rearrange the equation to find the minimum initial vertical velocity (Vy) required:
Vy = sqrt(2 * g * h)

Substituting the known values:
Vy = sqrt(2 * 9.8 * 6.5)

Calculating the value:
Vy ≈ 11.9 m/s

Since we only need the minimum speed to clear the canyon, the minimum speed required is equal to the horizontal speed (Vx) at takeoff:
V_min = Vx = 11.9 m/s

Therefore, the daredevil must have a minimum speed of 11.9 m/s in order to clear the canyon.