A daredevil decides to jump a canyon of width

8.4 m. To do so, he drives a motorcycle up an
incline sloped at an angle of 12.6

.
The acceleration of gravity is 9.8 m/s
2
.
What minimum speed must he have in order to clear the canyon?
Answer in units of m/s

for a projectile fired with velocity v at angle θ, the range is

r = (v^2 sin2θ)/g
8.4 <= v^2/9.8 sin12.6°
v^2 >= 8.4*9.8/.218
v^2 >= 377.61
v >= 19.4 m/s

To determine the minimum speed the daredevil must have in order to clear the canyon, we can use the principles of projectile motion. The key idea is that the motorcycle's initial speed and launch angle will determine the range, or horizontal distance, it can travel.

Here are the steps to find the minimum speed:

Step 1: Resolve the gravitational force into its components.
Since the motorcycle is on an incline, the gravitational force can be broken down into two components: one parallel to the incline (mg*sinθ) and one perpendicular to the incline (mg*cosθ), where m is the mass of the motorcycle and θ is the angle of the incline.

Step 2: Calculate the acceleration along the incline.
The acceleration along the incline can be determined by dividing the force parallel to the incline by the mass of the motorcycle. Therefore, the acceleration along the incline is: a = (mg*sinθ)/m = g*sinθ.

Step 3: Calculate the time taken to cross the canyon.
Using the kinematic equation, s = ut + (1/2)at^2, we know the initial velocity along the incline (u) is zero (since the motorcycle starts from rest) and the distance crossed (s) is the width of the canyon (8.4 m). Solving for t, we get: t = sqrt((2s)/a).

Step 4: Determine the horizontal distance covered by the motorcycle.
The horizontal distance (range) covered by the motorcycle can be found using the formula: R = ut = v*cosθ * t, where v is the initial speed of the motorcycle and θ is the angle of the incline. Since the motorcycle must clear the 8.4 m wide canyon, the range must be at least equal to this width.

Step 5: Rearrange the equation to solve for v.
Using the relationship from Step 4, we can rearrange the equation to solve for v: v = 8.4 / (sqrt((2 * 8.4) / (g * sinθ) * cosθ)).

Now, substituting the given values: g = 9.8 m/s^2 and θ = 12.6°:

v = 8.4 / (sqrt((2 * 8.4) / (9.8 * sin(12.6) * cos(12.6))) ≈ 20.4 m/s

Therefore, the daredevil must have a minimum speed of approximately 20.4 m/s to clear the canyon.