The current world record motorcycle jump is 77.0 meters set by Jason Renie. Assume that he left the take off ramp at 12.0 degrees above horizontal and that the take off and landing heights are the same. Neglecting air resistance,deteremine his take-off speed

To determine Jason Renie's take-off speed, we can use the principles of projectile motion. We'll use the fact that the horizontal range (distance traveled) of a projectile can be calculated using the equation:

Range = (initial velocity)^2 * sin(2θ) / gravity

Where:
- Range is the horizontal distance traveled by the projectile
- Initial velocity is the speed at take-off
- θ is the launch angle above horizontal
- gravity is the acceleration due to gravity

In this case, the range is given as 77.0 meters and the launch angle θ is given as 12.0 degrees. We need to find the initial velocity.

First, let's convert the launch angle from degrees to radians since trigonometric functions work with radians. We have:

θ (in radians) = θ (in degrees) * π / 180
θ (in radians) = 12.0 * π / 180
θ (in radians) = 0.2094 radians

Next, we rearrange the equation for range to solve for the initial velocity:

(initial velocity)^2 = Range * gravity / sin(2θ)
(initial velocity) = √(Range * gravity / sin(2θ))

Plugging in the given values, where gravity is approximately 9.8 m/s²:

(initial velocity) = √(77.0 * 9.8 / sin(2 * 0.2094))

Calculating this expression:

(initial velocity) = √(754.6 / sin(0.4188))
(initial velocity) = √(754.6 / 0.4076)
(initial velocity) = √1851.0367
(initial velocity) ≈ 43.01 m/s (rounded to two decimal places)

Therefore, Jason Renie's take-off speed is approximately 43.01 m/s.