A daredevil is dead set on jumping the Grand Canyon at its narrowest point (180 meters across). He has set up a ramp at 25 degrees and the ramp's release point is 20 meters above the rim of the canyon. He claims that his bike will be going 40m/s by the time he leaves the ramp. A corporate sponsor of the jump offers you a million dollars if you place a target on the other side and he hits your target. Where do you place your target? Other than increasing his speed, what advice would you offer the daredevil to increase his odds of landing the jump safely?
To determine where to place your target, we need to calculate the jump distance of the daredevil and find the point that corresponds to 180 meters across the narrowest point of the Grand Canyon.
First, let's break down the given information:
- The height of the ramp release point above the rim of the canyon is 20 meters.
- The angle of the ramp is 25 degrees.
- The bike's speed when leaving the ramp is 40 m/s.
To calculate the jump distance, we can use the projectile motion equations. The horizontal distance traveled (range) can be calculated using the formula:
Range = (velocity^2 * sin(2θ)) / g
where:
- velocity is the initial velocity (40 m/s, in this case)
- θ is the angle of launch (25 degrees)
- g is the acceleration due to gravity, approximately 9.8 m/s^2
Plugging in the values:
Range = (40^2 * sin(2 * 25)) / 9.8
Calculating this, we find that the range of the jump is approximately 205.4 meters. Since the narrowest point of the canyon is 180 meters across, we should place the target slightly beyond the calculated distance, ensuring it is within a reasonable range.
As for advice to increase the daredevil's odds of landing the jump safely, here are a few considerations:
1. Speed: Increasing the initial speed will allow for greater distance coverage. So, increasing the bike's speed would increase the chances of a successful jump.
2. Angle: While the given angle of 25 degrees is suitable, modifying it slightly based on calculations of maximum range might help optimize the jump.
3. Ramp Design: Ensuring the ramp's design is sound and allows for a smooth takeoff while providing enough upward force could improve the jump's outcome.
4. Safety Precautions: Utilize appropriate safety equipment, such as a well-fitted helmet, padding, and any other necessary gear to protect against potential impact or falls.
It's important to note that attempting such a dangerous feat comes with significant risks. The daredevil should consult with professionals, such as engineers or stunt experts, to evaluate and optimize all aspects of the jump to ensure their safety.