Please explain,
A clown in a circus is about to be shot out of a cannon with a muzzle velocity of 15.2 m/s, aimed at 52.7° above the horizontal. How far away should his fellow clowns position a net to ensure that he lands unscathed? The net is at the same height as the mouth of the cannon.
Thank you.
To determine the required distance for the net, we need to analyze the projectile motion of the clown. The initial velocity of the clown is given as 15.2 m/s, and the angle of projection is 52.7° above the horizontal.
Here are the steps to solve the problem:
Step 1: Split the initial velocity into its horizontal and vertical components. The horizontal component (Vx) remains constant throughout the motion, while the vertical component (Vy) changes due to the influence of gravity.
Vx = V * cos(θ)
Vy = V * sin(θ)
where V is the initial velocity (15.2 m/s) and θ is the angle of projection (52.7°).
Step 2: Determine the time it takes for the clown to reach the highest point of its trajectory. At the highest point, the vertical component of velocity is zero. We can use this information to find the time of flight (t).
Vy = V * sin(θ) - g * t
where g is the acceleration due to gravity (9.8 m/s²).
Step 3: Calculate the time of flight to determine the total time the clown is in the air.
t = (2 * Vy) / g
Step 4: Find the horizontal distance traveled by the clown using the equation:
Sx = Vx * t
where Sx is the horizontal distance traveled.
Step 5: The required distance for the net will be the horizontal distance traveled by the clown before landing. So, the final answer is Sx.
Now, let's calculate the solution:
Step 1:
Vx = 15.2 * cos(52.7°) = 9.85 m/s
Vy = 15.2 * sin(52.7°) = 11.57 m/s
Step 2:
0 = 11.57 - 9.8 * t
t = 11.57 / 9.8 = 1.179 s
Step 3:
t = (2 * 11.57) / 9.8 = 2.35 s
Step 4:
Sx = 9.85 * 2.35 = 23.11 m
Therefore, the net should be positioned approximately 23.11 meters away from the mouth of the cannon to ensure that the clown lands unscathed.