A daredevil decides to jump a canyon of

width 12 m. To do so, he drives a motorcycle
up an incline sloped at an angle of 14.2◦.
The acceleration of gravity is 9.8 m/s2 .
What minimum speed must he have in order
to clear the canyon?
Answer in units of m/s.

Range = Vo^2*sin(2A)/g = 12 m.

Vo^2*sin(28.4)/9.8 = 12
Vo^2*sin(28.4) = 117.6
Vo^2 = 117.6/sin(28.4) = 247.3
Vo = 15.72 m/s.

To determine the minimum speed required to clear the canyon, we can use the principles of projectile motion. Let's break down the problem into several steps:

Step 1: Resolve the gravitational acceleration into components.
Since the incline is sloped, we need to find the component of gravity acting along the incline. We can do this by finding the sine of the angle:

gsinθ = g * sin(14.2°)
= 9.8 m/s^2 * sin(14.2°)
= 2.53 m/s^2

Step 2: Calculate the time of flight.
The time it takes for the daredevil to travel a horizontal distance of 12 m can be calculated using the equation:

t = (2 * d) / v
where t is the time of flight, d is the horizontal distance, and v is the horizontal velocity.

In this case, the daredevil wants to reach the other side of the canyon, so the horizontal distance is equal to 12 m. The horizontal velocity is the same as the initial velocity.

Step 3: Determine the final vertical position.
Using the equation of motion:

y = V0y * t + (1/2) * g * t^2
where y is the vertical displacement, V0y is the vertical velocity at launch, t is the time of flight, and g is the gravitational acceleration.

Since the daredevil wants to clear the canyon, the vertical displacement should be greater than or equal to the width of the canyon. Thus:

y >= 12 m

Step 4: Find the initial vertical velocity.
We can find the vertical velocity at launch using the equation:

V0y = g * cos(θ)
= 9.8 m/s^2 * cos(14.2°)
= 9.41 m/s

Step 5: Substitute the values into the equation for the final vertical position.
Using the equation from step 3, we get:

y = V0y * t + (1/2) * g * t^2
12 >= (9.41 m/s) * t + (1/2) * (9.8 m/s^2) * t^2

Step 6: Solve for the minimum velocity.
Rearranging the equation from step 5 and solving for t, we get a quadratic equation:

(1/2) * (9.8 m/s^2) * t^2 + (9.41 m/s) * t - 12 = 0

Using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / 2a

where a = (1/2) * (9.8 m/s^2), b = (9.41 m/s), and c = -12, we can solve for t.

t = (-9.41 ± √((9.41)^2 - 4 * (1/2) * (9.8) * (-12))) / (2 * (1/2) * (9.8))
t ≈ 1.25 s

Step 7: Calculate the minimum velocity.
Using the equation from step 2, we can find the initial velocity (or the minimum speed required):

v = d / t
= 12 m / 1.25 s
= 9.6 m/s

Therefore, the daredevil must have a minimum velocity of approximately 9.6 m/s in order to clear the canyon.

To find the minimum speed the daredevil must have in order to clear the canyon, we can use the principles of projectile motion. Here's how you can approach this problem:

1. Break down the motion into horizontal and vertical components. The incline affects the initial speed and launch angle, but doesn't impact the horizontal motion.

2. Find the vertical component of the velocity required to clear the canyon. Since the daredevil wants to clear the canyon, the vertical distance he needs to cover is 12 meters. We can use the kinematic equation for vertical motion: Δy = v₀y * t + 0.5 * a * t², where Δy is the vertical distance, v₀y is the initial vertical velocity, a is the acceleration due to gravity, and t is the time of flight. Since the daredevil takes off and lands at the same height, the change in height Δy is zero. Solving the equation for v₀y, we get: v₀y = -0.5 * a * t.

3. Find the horizontal component of the velocity required. The horizontal component of the velocity remains constant throughout the motion. It is given by v₀x, which is the initial velocity along the incline. We can find v₀x using the formula v₀x = v₀ * cos(θ), where v₀ is the initial speed and θ is the angle of the incline.

4. The magnitude of the initial velocity can be found using the Pythagorean theorem: v₀ = √(v₀x² + v₀y²).

5. Rearrange the vertical component equation and substitute the value of v₀y to solve for t: 0 = -0.5 * a * t. By rearranging, we can find t = 0.

6. Finally, substitute the value of t = 0 into the horizontal component equation and solve for v₀x. This will give us the value of v₀x required to clear the canyon.

Let's plug in the given values into these equations:

a = 9.8 m/s² (acceleration due to gravity)
Δy = 12 m (vertical distance)
θ = 14.2° (angle of the incline)

First, find v₀x:
v₀x = v₀ * cos(θ)

Next, find v₀y:
Δy = v₀y * t + 0.5 * a * t²
Since Δy = 0 (no change in height), we get:
0 = -0.5 * a * t
Since t = 0, v₀y = 0.

Now, find v₀:
v₀ = √(v₀x² + v₀y²)

Substitute the values of v₀x and v₀y, and calculate v₀ to get the minimum speed required to clear the canyon.