You are launching Robin Hood out of a cannon and over the wall of the castle to rescue the Maid Marion. He would like to clear the castle wall by as much as possible. The cannon is stuck at an angle of 45 degrees from the ground and will launch him with a speed of 25 m/s. How close does he need to be to the wall to have the largest gap between him and the top of the wall as he flies over it?
the wall is on the axis of symmetry of the parabola
his initial vertical speed is
... 25 m/s * sin(45º)
... gravity slows him down to zero at his vertical peak
... the times is ... 25 * sin(45º) / 9.8
his horizontal speed is
... 25 m/s * cos(45º)
... use the time to peak to find the horizontal distance from the wall for maximum gap
To find out how close Robin Hood needs to be to the wall to have the largest gap between him and the top of the wall as he flies over it, we can use the concepts of projectile motion.
First, we need to identify the relevant equations of projectile motion. The most useful equation for our problem is the vertical displacement equation:
d = (vi^2 * sin^2θ) / (2g)
Where:
- d is the vertical displacement
- vi is the initial velocity (25 m/s)
- θ is the launch angle (45 degrees)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
However, the equation gives us the vertical displacement, which is the gap between Robin Hood and the top of the wall. We need to find the horizontal distance between Robin Hood and the wall.
To do that, let's use the equation for horizontal displacement:
dx = vi * cosθ * t
Where:
- dx is the horizontal displacement
- vi is the initial velocity (25 m/s)
- θ is the launch angle (45 degrees)
- t is the time of flight
Since we want Robin Hood to clear the top of the wall, we can consider the vertical displacement to be equal to the height of the wall.
Now, we need to determine the time of flight, which is the time it takes for Robin Hood to reach the maximum height and then descend back down to the ground level. In projectile motion, the time of flight is given by:
t = 2 * (vi * sinθ) / g
Now we have all the necessary equations to solve the problem.
1. Calculate the time of flight:
t = 2 * (25 m/s * sin(45 degrees)) / 9.8 m/s^2
2. Calculate the vertical displacement (gap between Robin Hood and the top of the wall):
d = (25 m/s)^2 * sin^2(45 degrees) / (2 * 9.8 m/s^2)
3. Calculate the horizontal displacement:
dx = 25 m/s * cos(45 degrees) * t
To find the minimum distance Robin Hood needs to be from the wall to have the largest gap over the top of the wall, we need to subtract the horizontal displacement from the gap:
distance_from_wall = dx - d
Solve these equations to find the answer.