find the angle of elevation of a gun that fires a shell with a muzzle velocity of 120m/s and hits a target on the same level but 1300m distant.
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To find the angle of elevation of the gun, we can use the concepts of projectile motion and trigonometry. Let's break down the problem step by step:
Step 1: Identify the given information:
- Muzzle velocity of the shell: 120 m/s
- Distance to the target: 1300 m
Step 2: Understand the projectile motion:
When an object is launched at an angle to the ground, it follows a curved trajectory known as projectile motion. The object's horizontal and vertical motions are independent of each other.
Step 3: Determine the time of flight:
Since the target is on the same level as the gun, the vertical displacement of the shell will be zero. Therefore, we can analyze only the horizontal motion. Let's calculate the time of flight using the formula:
time = distance / velocity
time = 1300 m / 120 m/s = 10.83 s (rounded to two decimal places)
Step 4: Calculate the initial vertical velocity:
To find the initial vertical velocity, we need to consider that the vertical displacement is zero and the time of flight is half of the total time spent by the shell in the air.
Using the formula:
Vertical displacement = (initial vertical velocity * time) + (0.5 * acceleration due to gravity * time^2)
Since the vertical displacement is zero, we can rewrite the formula as:
0 = (initial vertical velocity * (10.83 / 2)) - (0.5 * 9.8 * (10.83 / 2)^2)
Simplifying the equation:
initial vertical velocity * 5.415 = 0.5 * 9.8 * 5.415^2
initial vertical velocity = (0.5 * 9.8 * 5.415^2) / 5.415
initial vertical velocity ≈ 25.15 m/s (rounded to two decimal places)
Step 5: Calculate the angle of elevation:
Considering the projectile motion, we have the horizontal and vertical components of the initial velocity. The angle of elevation (θ) can be obtained using the tangent function:
Tangent(θ) = (initial vertical velocity) / (muzzle velocity)
Tangent(θ) = 25.15 m/s / 120 m/s
Taking the arctangent (inverse tangent) of both sides:
θ = arctan(25.15 / 120)
Using a calculator or mathematical software, we find that θ ≈ 11.94 degrees (rounded to two decimal places).
Therefore, the angle of elevation of the gun is approximately 11.94 degrees.