Question

You have been employed by the local circus to plan their human cannonball performance. For this act, a spring-loaded cannon will shoot a human projectile, the Great Flyinski, across the big top to a net below. The net is located \(5.0 \mathrm{m}\) lower than the muzzle of the cannon from which the Great Flyinski is launched. The cannon will shoot the Great Flyinski at an angle of \(35.0^{\circ}\) above the horizontal and at a speed of $18.0 \mathrm{m} / \mathrm{s} .$ The ringmaster has asked that you decide how far from the cannon to place the net so that the Great Flyinski will land in the net and not be splattered on the floor, which would greatly disturb the audience. What do you tell the ringmaster? ( Wheractive: projectile motion)

Short answer

Answer: To find the distance the net should be placed, first calculate the horizontal and vertical components of the initial velocity. Then, find the time of flight using the vertical component and the height difference between the cannon and the net. Finally, use the time of flight and horizontal component to calculate the horizontal range where the net should be placed.

Step-by-step solution

Calculate horizontal and vertical components of velocity.

First, we need to find the horizontal (u_x) and vertical (u_y) components of the initial velocity of the Great Flyinski. Use the initial velocity (18 m/s) and the launch angle (35°) to find these components: u_x = u * cos(35°) u_y = u * sin(35°)

Find time of flight.

Next, we'll find the time for which the Great Flyinski is in the air (t_flight) by using the vertical component of velocity. We know the vertical displacement (Δy) which is `5.0 m` (the net is lower than the muzzle of the cannon) and we can use the formula for constant acceleration due to gravity (g = 9.8 m/s²) to find the time: Δy = u_y * t_flight - (1/2) * g * t_flight²

Calculate the horizontal range.

Now, we will use the time of flight (t_flight) and horizontal component of velocity (u_x) to find the horizontal distance (R) the net should be placed: R = u_x * t_flight

Solve for t_flight in Step 2.

Rearrange the equation obtained in Step 2 to isolate t_flight and then solve for it in terms of u_y, g, and Δy: t_flight = (u_y + sqrt(u_y² - 2 * g * Δy)) / g, or t_flight = (u_y - sqrt(u_y² - 2 * g * Δy)) / g

Find the value of R.

Substitute the values for u_x and t_flight obtained in the previous steps into the equation for R from Step 3: R = u_x * ((u_y + sqrt(u_y² - 2 * g * Δy)) / g) or R = u_x * ((u_y - sqrt(u_y² - 2 * g * Δy)) / g) Finally, we will calculate the approximate value for R. The ringmaster should place the net at this distance from the cannon to ensure that the Great Flyinski lands safely in the net.