An artillery shell is fired at an angle of 56.7 degrees
above the horizontal ground with an initial
speed of 1960 m/s. The acceleration of gravity is 9.8 m/s^2.
Find the total time of flight of the shell and the horizontal range,
neglecting air resistance.
Vo = 1960m/s[56.7o].
Xo = 1960*Cos56.7 = 1076 m/s
Yo = 1960*sin56.7 = 1638 m/s.
Y = Yo + g*Tr = 0.
1638 + (-9.8)Tr = 0,
Tr = 167.2 s. = Rise time.
Tf = Tr = 167.2 s. = Fall time.
T = Tr + Tf = 167.2 + 167.2 = 334.4 s. = Time in flight.
Range = Xo * T = 1076 * 334.4 =
To find the total time of flight and the horizontal range of the artillery shell, we can use the following equations of motion:
1. The equation for the horizontal range (R):
R = (v₀ * cosθ) * t
2. The equation for the vertical motion (h):
h = (v₀ * sinθ) * t - (0.5 * g * t²)
3. The equation for the total time of flight (T):
T = 2 * t
where:
- v₀ is the initial speed of the shell (1960 m/s).
- θ is the angle of elevation (56.7 degrees).
- g is the acceleration due to gravity (9.8 m/s²).
- t is the time it takes for the shell to reach the ground.
Now, let's calculate the horizontal range and the total time of flight step by step:
Step 1: Calculate the horizontal range (R)
Using equation (1), substitute v₀ = 1960 m/s and θ = 56.7 degrees:
R = (1960 m/s * cos(56.7 degrees)) * t
Step 2: Calculate the total time of flight (T)
Using equation (3), T = 2 * t
Step 3: Find the time of flight (t)
To find the time of flight, we need to determine the time when the vertical motion ends. This occurs when the shell hits the ground, meaning the height (h) is equal to zero.
Using equation (2), substitute h = 0 and solve for t:
0 = (1960 m/s * sin(56.7 degrees)) * t - (0.5 * 9.8 m/s² * t²)
This equation is a quadratic equation in terms of t. Solve it to find the value of t.
Step 4: Substitute the value of t into equation (1) to find the horizontal range (R).
By following these steps, you can find the total time of flight and the horizontal range of the artillery shell.