A daredevil decides to jump a canyon. It's walls are equally high and 13 m apart. He takes off by driving a motorcycle up a short ramp sloped at an angle of 18°. What minimum speed must he have in order to clear the canyon?
Awesome generated quite q bit of discussion. :)
To determine the minimum speed required for the daredevil to clear the canyon, we can break down the problem into two components: the horizontal distance and the vertical height.
1. Horizontal Distance:
The canyon walls are 13 meters apart. Since the daredevil needs to clear the canyon, we can consider this distance as the horizontal distance traveled during the jump.
2. Vertical Height:
The daredevil launches off a ramp sloped at an angle of 18 degrees. We can use this angle to calculate the vertical height required to clear the canyon.
Now, let's calculate the vertical height using trigonometry. We can use the sine function (sin) with the angle and the horizontal distance.
sin(18°) = vertical height / 13 m
Rearranging the equation, we get:
vertical height = sin(18°) * 13 m
Now, we have both the horizontal distance and the vertical height. To calculate the minimum speed needed, we can use the kinematic equation for projectile motion:
V² = U² + 2as
Where:
V = final velocity (unknown)
U = initial velocity (minimum speed needed)
a = acceleration (assumed to be g, gravitational acceleration = 9.8 m/s²)
s = displacement (vertical height)
Since the daredevil takes off from the ramp, his initial vertical velocity (U) would be 0 m/s.
V² = 0² + 2 * 9.8 * vertical height
Now, let's substitute the value of the vertical height and solve for V:
V² = 2 * 9.8 * sin(18°) * 13
V² = 254.809
Taking the square root of both sides:
V = √254.809
V ≈ 15.98 m/s
Therefore, the daredevil must have a minimum speed of approximately 15.98 m/s in order to clear the canyon.
To determine the minimum speed required for the daredevil to clear the canyon, we can use the concept of projectile motion. We can break down the daredevil's motion into horizontal and vertical components.
First, let's analyze the vertical component. The daredevil wants to clear a 13 m wide canyon, so the vertical distance he needs to cover to clear the canyon walls is also 13 m.
The vertical distance can be calculated using the equation for projectile motion:
Δy = v₀y * t + (1/2) * g * t²
Where:
Δy = vertical distance (13 m)
v₀y = initial vertical velocity
t = time of flight
g = acceleration due to gravity (-9.8 m/s²)
Since the daredevil wants to clear the canyon entirely, his initial vertical velocity (v₀y) should be zero at the highest point of his trajectory. This means that the vertical distance equation simplifies to:
Δy = (1/2) * g * t²
Now, let's analyze the horizontal component. The daredevil will be driving up a ramp sloped at an angle of 18°. The horizontal distance he travels can be calculated using the equation:
Δx = v₀x * t
Where:
Δx = horizontal distance (13 m)
v₀x = initial horizontal velocity
t = time of flight
Since the daredevil wants to clear the canyon entirely, the horizontal distance (Δx) should be equal to the width of the canyon (13 m). Therefore:
13 m = v₀x * t
Now, to find the minimum speed required, we need to find the time of flight (t) in both equations. We can solve the second equation for t:
t = 13 m / v₀x
Now we substitute this value of t into the first equation and solve for v₀x:
13 m = v₀x * (13 m / v₀x) = (1/2) * g * (13 m / v₀x)²
Simplifying the equation:
13 = (1/2) * g * (13 / v₀x)²
26 = g * (169 / v₀x²)
26 * v₀x² = 169 * g
v₀x² = (169 * g) / 26
v₀x = √ [(169 * g) / 26]
Plugging in the value of g (acceleration due to gravity) as 9.8 m/s²:
v₀x = √ [(169 * 9.8) / 26] ≈ 9.99 m/s
Therefore, the daredevil must have a minimum horizontal velocity (speed) of approximately 9.99 m/s in order to clear the canyon.
Use range equation:
d = [v^2(sin2theta)]÷g
18 = [v^2(0.58778)]÷9.8
(18 x 9.8)÷0.58778 = v^2
v^2 = 216.745
v = 14.722 m/s
Hope this helps :)