The range of a target is found to be 20km. A shell leaves a gun with a velocity of 500m/s. What must be the angle of elevation of the gun. If the ground is level
The range of a projectile launched at velocity V and an angle θ to the horizontal is:
R = (V^2 Sin2θ)/g
Here, R = 20km = 20,000m and V = 500m/s, so
20,000m x 9.8m/s^2 = 500^2 (m/s)^2 • Sin 2θ
So, Sin2θ = 0.784
θ = 0.4585 radians = 25.812°
The angle of the gun is 25.812 degrees to the horizontal.
I need the solution
Recall that the range
R = v^2/g sin2θ
so you just need to find θ such that
500^2/9.81 sin2θ = 20000
To find the angle of elevation of the gun, we need to use the kinematic equations of motion.
First, we can break down the initial velocity of the shell into its horizontal and vertical components. Since the ground is level, the horizontal component remains constant throughout the motion, while the vertical component changes due to gravity.
The horizontal component of the velocity (Vx) is given by:
Vx = V * cos(θ)
where V is the initial velocity (500 m/s) and θ is the angle of elevation.
The vertical component of the velocity (Vy) is given by:
Vy = V * sin(θ)
Next, let's find the time it takes for the shell to reach the target. We can use the equation of motion for vertical displacement:
y = Vy * t - (1/2) * g * t^2
where y is the vertical displacement (0 because the ground is level) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we get:
0 = V * sin(θ) * t - (1/2) * g * t^2
Since the range of the target is 20 km (20,000 m), we can use the equation of motion for horizontal displacement:
x = Vx * t
where x is the horizontal displacement (20,000 m).
Plugging in the values, we get:
20,000 = V * cos(θ) * t
Now we have two equations with two unknowns (θ and t). We can solve these equations simultaneously to find the angle of elevation (θ).
Dividing the second equation by the first equation:
(20,000) / (V * cos(θ) * t) = (V * sin(θ) * t) / ((1/2) * g * t^2)
Simplifying:
40,000 / (V * cos(θ)) = (V * sin(θ)) / (1/2 * g * t)
Canceling out t:
40,000 / (V * cos(θ)) = 2 * (V * sin(θ)) / (g)
Rearranging the equation:
40,000 * g = 2 * V^2 * sin(θ) * cos(θ)
Using the identity sin(2θ) = 2 * sin(θ) * cos(θ):
40,000 * g = V^2 * sin(2θ)
Finally, solving for θ:
θ = (1/2) * sin^(-1)((40,000 * g) / V^2)
Plugging in the values:
θ = (1/2) * sin^(-1)((40,000 * 9.8) / (500^2))
Using a calculator, we can evaluate the expression to find the angle of elevation (θ).