Find the angle of elevation of a gun that fires a shell with muzzle velocity of 125m/s and hits a target on the same level but 1.55km distance
To find the angle of elevation of the gun, we can use the concept of projectile motion.
First, let's break down the initial velocity into its horizontal and vertical components.
The horizontal component of velocity remains constant throughout the projectile's motion. Therefore, the horizontal velocity (Vx) can be found using the formula:
Vx = initial velocity * cos(angle)
where the angle is the angle of elevation.
In this case, the initial velocity is 125 m/s and we want to find the angle of elevation. Let's call it θ:
Vx = 125 m/s * cos(θ)
Next, let's calculate the time of flight (t):
The time of flight for a projectile can be calculated using the formula:
t = 2 * (initial vertical velocity) / acceleration due to gravity
Since the projectile is fired on the same level and lands at the same level, the vertical displacement is zero. Hence, the initial vertical velocity is zero.
t = 2 * (0) / 9.8 m/s^2 = 0 seconds
Since the vertical displacement is zero, the time of flight is zero.
Now, let's calculate the horizontal displacement (range) using the formula:
range = horizontal velocity * time of flight
In this case, the range is 1.55 km (1550 m) and the time of flight is 0 s:
1.55 km = Vx * 0
0 = Vx
Since Vx = 125 m/s * cos(θ), we can conclude that cos(θ) = 0.
The cosine of an angle is equal to zero when the angle is 90 degrees (π/2 radians).
Therefore, the angle of elevation of the gun that fires the shell is 90 degrees (π/2 radians).
To find the angle of elevation of the gun, we can use the principles of projectile motion. We need to calculate the vertical distance the shell travels and the horizontal distance it travels.
Step 1: Calculate the time of flight
Using the vertical motion equation:
h = v0y * t - (1/2) * g * t^2
where:
h = 0 (since the shell hits the target at the same level)
v0y = vertical component of the initial velocity (muzzle velocity * sinθ)
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time of flight
Plugging in the known values, we get:
0 = (125 * sinθ) * t - (1/2) * 9.8 * t^2
Step 2: Calculate the horizontal distance
Using the horizontal motion equation:
d = v0x * t
where:
d = horizontal distance
v0x = horizontal component of the initial velocity (muzzle velocity * cosθ)
t = time of flight (calculated in Step 1)
Plugging in the known values, we get:
1550 m = (125 * cosθ) * t
Step 3: Solve for the angle of elevation
We have two equations with two unknowns (θ and t). We can solve these equations simultaneously to find the value of θ.
From the second equation, we can isolate t:
t = 1550 / (125 * cosθ)
Substituting this value of t into the first equation, we get:
0 = (125 * sinθ) * (1550 / (125 * cosθ)) - (1/2) * 9.8 * (1550 / (125 * cosθ))^2
Simplifying this equation, we get:
0 = 2 * sinθ * (1550 / cosθ) - (9.8 * 1550^2) / (125^2 * cos^2θ)
Simplifying further, we can rewrite this equation as:
2 * sinθ * cosθ = (9.8 * 1550^2) / (125^2)
Using the identity sin2θ = 2 * sinθ * cosθ, we can rewrite the equation as:
sin2θ = (9.8 * 1550^2) / (125^2)
Now, we can solve for θ by taking the inverse sine (arcsin) of both sides of the equation:
θ = arcsin[((9.8 * 1550^2) / (125^2))^0.5]
Calculating this expression will give you the angle of elevation (θ) of the gun.
the range formula is
d= (v^2 / g) sin(2Θ)