A daredevil jumps a canyon 10.6 m wide. To do so, he drives a car up a 19° incline. (The daredevil lands on the other side of the canyon at the same elevation as takeoff.)
What minimum speed must he achieve to clear the canyon?
To determine the minimum speed the daredevil must achieve to clear the canyon, we can use the principles of projectile motion. Let's break down the problem into components:
1. Determine the initial velocity in the horizontal direction (x-direction).
Since the daredevil wants to clear the canyon horizontally, the initial velocity in the x-direction will be the horizontal component of the car's velocity. We can calculate it using the given angle of the incline and the total initial velocity (V).
Vx = V * cosθ, where Vx is the initial horizontal velocity, V is the total initial velocity, and θ is the angle of the incline.
2. Determine the initial velocity in the vertical direction (y-direction).
The initial vertical velocity of the car can be calculated using the given angle of the incline and the total initial velocity (V).
Vy = V * sinθ, where Vy is the initial vertical velocity, V is the total initial velocity, and θ is the angle of the incline.
3. Calculate the time the car takes to reach the other side of the canyon.
Since the daredevil wants to land on the other side of the canyon, the time it takes for the car to traverse the canyon horizontally will be the same as the time it takes for the car to achieve its vertical displacement. Therefore, we can use the vertical displacement and the initial vertical velocity to calculate the time (t).
Δy = Vy * t - 0.5 * g * t^2, where Δy is the vertical displacement, Vy is the initial vertical velocity, t is the time, and g is the acceleration due to gravity (-9.8 m/s^2).
4. Calculate the horizontal displacement during that time.
The horizontal displacement of the car can be calculated using the horizontal velocity (Vx) and the time (t) calculated in the previous step.
Δx = Vx * t, where Δx is the horizontal displacement, Vx is the initial horizontal velocity, and t is the time.
5. Set the horizontal displacement equal to the width of the canyon.
Since the car needs to clear the canyon, we set the horizontal displacement equal to the width of the canyon and solve for the initial horizontal velocity (Vx).
Δx = 10.6 m,
Vx * t = 10.6 m,
Vx * (Δy / Vy) = 10.6 m,
Vx = (10.6 m * Vy) / Δy.
6. Substitute the expressions for Vy and Δy.
Substitute the expressions from earlier into the equation for Vx to solve for the initial horizontal velocity.
Vx = (10.6 m * V * sinθ) / (Vy * (Vy / g)),
Vx = (10.6 m * V * sinθ) / (2 * g * Δy).
7. Find the minimum initial velocity V.
Finally, to find the minimum initial velocity, we set V equal to the total initial velocity V.
V = Vx / cosθ,
V = ((10.6 m * V * sinθ) / (2 * g * Δy)) / cosθ,
V = (10.6 m * V * sinθ) / (2 * g * Δy * cosθ),
V = (10.6 m * V * sinθ) / (2 * g * Δy * cosθ) + (10.6 m * V * sinθ).
Rearranging this equation to solve for V:
V - (10.6 m * V * sinθ) / (2 * g * Δy * cosθ) = (10.6 m * V * sinθ),
V - (10.6 m * V * sinθ) = (10.6 m * V * sinθ) * (2 * g * Δy * cosθ),
V * (1 - 10.6 m * sinθ) = 2 * g * Δy * cosθ,
V = (2 * g * Δy * cosθ) / (1 - 10.6 m * sinθ).
After solving this equation, you will obtain the minimum initial velocity (V) the daredevil must achieve to clear the canyon.