A fighter plane is directly over an anti aircraft gun at a time t=0. The plane has a speed of 600km/hr. A shell is fired from an anti aircraft gun at the time t=0, in order to hit the plane. The muzzle velocity V is 1100m/s. Determine the distance the gun should be aimed ahead of the plane in order to hit it if the distance of the plane from the ground is 2500m.

Question

A fighter plane is directly over an anti aircraft gun at a time t=0. The plane has a speed of 600km/hr. A shell is fired from an anti aircraft gun at the time t=0, in order to hit the plane. The muzzle velocity V is 1100m/s. Determine the distance the gun should be aimed ahead of the plane in order to hit it if the distance of the plane from the ground is 2500m.

Answer 1

the plane and the shell cover the same horizontal distance in time t

T = angle up from horizontal

u = 1100 cos T = 600,000/3600 m/s

so T = 81.3 degrees

so Vi = initial speed up = 1100sin 81.3
= 1087 m/s up

how long to reach 2500 meters high?
2500 = 1087 t - 4.9 t^2

4.9 t^2 + 1087 t - 2500 = 0

solve for t, use + result

d = u t

Answer 2

To determine the distance the gun should be aimed ahead of the plane, we need to consider the horizontal motion of both the plane and the shell.

Let's break down the problem into two components: horizontal (x-axis) and vertical (y-axis). The horizontal motion will determine the distance the gun should be aimed ahead, while the vertical motion will determine the time it takes for the shell to reach the plane's altitude.

First, let's determine the time it takes for the shell to reach the plane's altitude of 2500 meters. We'll use the vertical motion equation:

h = V₀t + (1/2)gt²

Where:
h = altitude of the plane = 2500 m
V₀ = initial vertical velocity = 0 m/s (the shell is fired horizontally)
g = acceleration due to gravity = 9.8 m/s²
t = time taken to reach the altitude

Rearranging the equation, we get:

t = √(2h/g)

Substituting the given values:

t = √(2 * 2500 / 9.8) = √(5000 / 9.8) ≈ 10.20 seconds

Now that we have the time it takes for the shell to reach the altitude, we can calculate the distance the plane travels horizontally during this time. We'll use the horizontal motion equation:

d = V × t

Where:
d = horizontal distance traveled by the plane
V = horizontal velocity of the plane = 600 km/hr = (600 * 1000) / 3600 m/s (convert km/hr to m/s)
t = time taken for the shell to reach the altitude (calculated earlier)

Converting the velocity:

V = (600 * 1000) / 3600 ≈ 166.67 m/s

Substituting the values:

d = 166.67 m/s × 10.20 s ≈ 1700.01 meters

So, the distance the gun should be aimed ahead of the plane in order to hit it is approximately 1700 meters.