A daredevil decides to jump a canyon of width
14.6 m. To do so, he drives a motorcycle up
an incline sloped at an angle of 20.8
◦
.
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.
To find the minimum speed the daredevil must have to clear the canyon, we can start by analyzing the forces acting on the motorcycle.
The main forces to consider are the gravitational force (mg) pulling the motorcycle downwards and the normal force (N) exerted by the incline to support the motorcycle's weight. The normal force can be resolved into two components: perpendicular to the incline (N⊥) and parallel to the incline (N‖).
Let's draw a force diagram for the motorcycle on the incline:
```
|\
| \
| \
| \
| \
N⊥ | \
------|-------
| \
| mg
|______\
N‖
```
Here, mg represents the weight of the motorcycle acting vertically downwards, N⊥ represents the perpendicular component of the normal force acting upwards, and N‖ represents the parallel component of the normal force acting along the incline.
Since we are interested in finding the minimum speed, we can neglect friction as it would only slow down the motorcycle. Therefore, the net force acting on the motorcycle should be equal to zero in the horizontal direction.
The net force in the horizontal direction is given by the component of the gravitational force parallel to the incline (mg*sinθ) minus the force required to accelerate the motorcycle up the incline (ma):
net force = mg*sinθ - ma
Since the net force is zero, we have:
mg*sinθ - ma = 0
Rearranging the equation, we find:
ma = mg*sinθ
The acceleration of the motorcycle up the incline is related to the minimum speed (v) by the equation a = v² / (2s), where s is the distance traveled on the incline.
The distance traveled on the incline can be found using the distance formula s = ut + (1/2)at², where u is the initial velocity (0 m/s) and t is the time taken to travel the distance s.
Since the motorcycle starts from rest (u = 0 m/s), the equation simplifies to s = (1/2)at².
Plugging this expression for s into the equation for acceleration a = v² / (2s), we get:
a = v² / [2 * (1/2)at²]
Simplifying, we have:
a = v² / t²
Now we can substitute this expression for a in the equation ma = mg*sinθ:
v² / t² = g*sinθ
Rearranging the equation further, we find:
v² = g*sinθ * t²
To clear the canyon, the daredevil should have enough speed to cover the width of the canyon, which is 14.6 m. Since s = vt, we can substitute this equation into the equation above:
v² = g*sinθ * s / t
Now we have a relationship between the minimum speed (v), acceleration due to gravity (g), angle of the incline (θ), distance across the canyon (s), and the time taken to cross the canyon (t).
To solve for the minimum speed, we need to find the value of t. The time taken to cross the canyon is given by:
t = s / (v*cosθ)
Substituting this expression for t back into the equation for v², we get:
v² = g*sinθ * s / (s / (v*cosθ))
Simplifying, we find:
v² = g*sinθ * v*cosθ
Now we can solve for v:
v² = (g*sinθ * v*cosθ)
v² = g*sinθ * v*cosθ
Dividing both sides by v*cosθ, we get:
v = g*sinθ
Plugging in the given values: g = 9.8 m/s² and θ = 20.8°, we can now solve for v:
v = 9.8 m/s² * sin(20.8°)
Using a calculator, we find:
v ≈ 3.423 m/s
Therefore, the daredevil must have a minimum speed of approximately 3.423 m/s to clear the canyon.
recall that the range of such a trajectory is
R = v^2/g sin2θ
You have θ and g, so set R=14.6 and solve for v