A daredevil decides to jump a canyon of width 4.22 m. To do so, he drives a motorcycle up an incline sloped at an angle of 12.4◦. The acceleration of gravity is 9.8 m/s. What minimum speed must he have in order to clear the canyon? Answer in units of m/s.
Range = Vo^2*sin(2A)/g.
4.22 = Vo^2*sin(24.8)/9.8.
4.22 = Vo^2*0.0428, Vo^2 = 98.6, Vo = 9.93 m/s.
To solve this problem, we can use the principle of projectile motion. The daredevil's motorcycle can be treated as a projectile launched at an angle of 12.4° with an initial velocity.
Let's break down the problem into two components: one along the incline and the other perpendicular to it.
1. Perpendicular Component:
The daredevil needs to clear the vertical height of the canyon, which is given by H = 0 m (the daredevil is on the same level at both ends of the canyon). The initial upward velocity (V_y) can be calculated using the equation V_y^2 = u^2 + 2as, where u is the initial vertical velocity, a is the acceleration (gravity), and s is the vertical distance traveled.
Since the daredevil starts from rest, u = 0. Therefore, we have V_y^2 = 0 + 2(-9.8)(0) = 0. The daredevil does not need any initial vertical velocity to clear the canyon since H = 0.
2. Along the Incline Component:
The daredevil needs to travel the horizontal distance (D) of the canyon, which is given by D = 4.22 m. We can calculate the minimum speed (V) required using the equation D = V_x * t, where V_x is the horizontal velocity and t is the time.
The horizontal velocity (V_x) can be calculated using the equation V_x = u * cos(θ), where u is the initial velocity, and θ is the angle of inclination.
Substituting the given values, we have V_x = u * cos(12.4°).
Now, we can calculate the time (t) taken to cover the horizontal distance using the equation D = V_x * t.
Substituting the known values, we have 4.22 = u * cos(12.4°) * t.
We need to find the minimum speed, so we want to find the minimum value of u. Therefore, we want to find the minimum value of t.
Rearrange the equation to solve for t:
t = 4.22 / (u * cos(12.4°)).
For the daredevil to clear the canyon, the time taken to cover the horizontal distance should be equal to or greater than the time taken for him to fall vertically. Therefore, t should be greater than or equal to the time taken for vertical fall, which is given by t = 2 * V_y / g.
Substituting V_y = 0, we have t = 2 * 0 / 9.8 = 0.
So, substituting t = 0 in the previous equation, we get:
0 ≤ 4.22 / (u * cos(12.4°)).
Simplifying the equation, we have 0 ≤ 4.22 / (u * cos(12.4°)).
To satisfy this equation, the denominator (u * cos(12.4°)) should be positive or greater than zero. Therefore, u should be greater than zero.
Hence, the minimum speed the daredevil must have in order to clear the canyon is any positive value greater than zero.
To find the minimum speed the daredevil must have in order to clear the canyon, we can use the principles of physics and apply Newton's Laws of Motion.
To begin, we need to break down the given information:
- Width of the canyon (d) = 4.22 m
- Angle of the incline (θ) = 12.4°
- Acceleration due to gravity (g) = 9.8 m/s²
We'll assume there are no external forces acting on the daredevil's motorcycle except for gravity. Therefore, we'll consider the forces along the inclined plane.
The forces acting on the motorcycle are:
1. Gravity force (mg), which acts vertically downwards and has two components:
- Normal force (N), perpendicular to the incline plane
- Component force parallel to the incline plane (mg*sin(θ))
2. Friction force (Ff), which opposes the motion and acts parallel to the incline plane
3. Net force (Fnet), which gives the motorcycle its acceleration
Now, we can analyze the forces in the horizontal and vertical directions using Newton's second law:
In the horizontal direction:
- Fnet = Ff
In the vertical direction:
- N - mg*cos(θ) = 0 (since the motorcycle is not sinking into the incline)
To solve for the minimum speed the daredevil must have, we need to determine the point where the motorcycle takes off from the incline. At this point, the motorcycle leaves the incline, and the normal force becomes zero. Let's call this point A.
When the normal force becomes zero (N = 0), we can write the equation:
0 - mg*cos(θ) = 0
mg*cos(θ) = 0
Simplifying the equation, we get:
mg*cos(θ) = 0
m*g = 0 (since cos(θ) ≠ 0 for any value of θ)
Since m and g are both positive values, this equation is not possible. It means that the motorcycle cannot leave the incline at point A since it will always experience a non-zero normal force.
Therefore, we conclude that the daredevil cannot clear the canyon using this incline at any speed.