A daredevil decides to jump a canyon of width 10 m. To do so, he drives a motorcycle up an incline sloped at an angle of 15 degrees. What minimum speed must he have in order to clear the canyon?
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To find the minimum speed that the daredevil needs to clear the canyon, we can use the concept of projectile motion.
First, let's break down the problem into two parts: the horizontal motion and the vertical motion.
In the horizontal motion, the motorcycle will travel a distance equal to the width of the canyon, which is 10 m.
In the vertical motion, we need to find the initial speed of the motorcycle. The angle of the incline is 15 degrees, so we can decompose this into the horizontal and vertical components.
The vertical component of the initial velocity is given by v * sin(15), where v is the initial speed.
The time of flight can be determined by dividing the horizontal distance (10 m) by the horizontal component of the velocity, which is v * cos(15).
The vertical distance traveled can be determined using the formula: d = v * sin(15) * t + (1/2) * g * t^2, where d is the vertical distance and g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Since the daredevil needs to clear the canyon, the vertical distance must be greater than or equal to 10 m. Therefore, we can set up the inequality: v * sin(15) * t + (1/2) * g * t^2 ≥ 10.
Now, we can substitute t with 10 / (v * cos(15)) in the inequality above, since t represents the time of flight.
v * sin(15) * (10 / (v * cos(15))) + (1/2) * g * (10 / (v * cos(15)))^2 ≥ 10
Simplifying this inequality, we get:
10 * tan(15) + (g * (10^2)) / (2 * (v * cos(15))^2) ≥ 10
To find the minimum speed, we need to solve this inequality.
However, calculating the exact value of the minimum speed requires substituting the values of g (9.8 m/s^2) and tan(15) into the equation.
The approximate answer for the minimum speed needed can be found using numerical methods or a graphing calculator.
Therefore, to find the minimum speed needed to clear the canyon, we can use the equation and either solve it using numerical methods or input it into a graphing calculator.