Question

Your friend has decided to make some money during the next State Fair by inventing a game of skill. In the game as she has developed it so far, the customer shoots a rifle at a 5.0 cm diameter target falling straight down. Anyone who hits the target in the center wins a stuffed animal. Each shot would cost 50 cents. The rifle would be mounted on a pivot 1.0 meter above the ground so that it can point in any direction at any angle. When shooting, the customer stands 100 meters from where the target would hit the ground if the bullet misses. At the instant that the bullet is fired (with velocity of 1200 ft/sec according to the manual), the target is released from its holder 7.0 meters above the ground. (You see, the trigger is electronically connected to the release mechanism.) Your friend asks you to try out the game which she has set up on a farm outside of town. Before you fire the gun you calculate where you should aim. You may, as usual, neglect any effects of air resistance. Your conclusion? At what angle from the vertical should you aim?

Answer 1

Answer: from the vertical, one should aim  86.6°

Explanation:

height of the center of object = 7.0 m - 0.05 m = 6.95 m

now let the bullet hits centre at point A height x meters from the ground

also let t be the time taken for the bullet to hit the object

so distance travelled by the target will be

d = h - x = 6.95 - x

now using the equation of motion

d = 1/2gt²

so 1/2gt² = 6.95 - x

x = 6.95 - 1/2gt² .........let this be equ 1

let angle of fire be ∅

so v(cos∅) × t = 100

our velocity v is 1200 ft/sec = 365.76 m/s

365.76(cos∅) × t = 100 ........equ 2

also vertical position of the bullet after t is

y = y₀ + c(sin∅)t - 1/2gt²

y = 1 + 365.76(sin∅)t - 1/2gt² ----- equ 3

After time t. the vertical position x and y are same, else the bullet wouldn't have strike target at centre, so;

x = y

we substitute

equ 1 = equ 3

6.95 - 1/2gt² = 1 + 365.76(sin∅)t - 1/2gt²

6.95 - 1 = 365.76(sin∅)t - 1/2gt² +  1/2gt²

5.95 = 365.76(sin∅)t

t = 5.96 / 365.76(sin∅)

now input the above equ  into equ 2

365.76(cos∅) ×  5.96/365.76(sin∅) = 100

5.95(cos∅)/sin∅ = 100

tan∅ = 5.95/100 = 0.0595

∅ =  3.40°

therefore from the vertical, one should aim (90° - 3.40°) = 86.6°