An artillery shell is fired at an angle of 67.4◦ above the horizontal ground with an initial speed of 1860 m/s.

The acceleration of gravity is 9.8 m/s2 .
Find the total time of flight of the shell, neglecting air resistance.
Answer in units of min

Vo = 1860m/s[67.4o]

Xo = 1860*Cos67.4 = 714.8 m/s.
Yo = 1860*sin67.4 = 1717.2 m/s.

Y = Yo-g*Tr = 0
1717.2 - 9.8Tr = 0
9.8Tr = 1717.2
Tr = 175 s. = Rise time.

Tf = Tr = 175 s. = Fall time

Tr+Tf = 175 + 175 = 350 s. = 5.83 Min. in flight.

To find the total time of flight of the shell, we can use the equations of projectile motion.

First, let's consider the vertical component of the shell's motion. We can use the equation for vertical displacement to find the time it takes for the shell to reach its maximum height:

Δy = V0y*t - (1/2)*g*t^2

Where:
Δy = vertical displacement (in this case, the maximum height)
V0y = initial vertical velocity (in this case, the initial speed * sin(angle))
t = time
g = acceleration due to gravity

Since the shell reaches its maximum height when it stops moving vertically (when its vertical velocity becomes zero), we can set V0y*t - (1/2)*g*t^2 equal to zero and solve for t:

0 = (initial speed * sin(angle)) * t - (1/2) * (acceleration due to gravity) * t^2

The above equation is a quadratic equation, and we can solve it by using the following formula:

t = (V0y ± √(V0y^2 + 2*g*Δy)) / g

Since we are interested in the time the shell takes to reach its maximum height, we take the positive root of the quadratic equation.

Now, let's calculate Δy (the maximum height). At the maximum height, the vertical velocity becomes zero. Therefore, we can use the equation for vertical velocity:

Vf = V0y - g*t

Setting Vf to zero, we can solve for t:

V0y = g*t
t = V0y / g

Now that we have the time it takes the shell to reach its maximum height, we can use this information to find the total time of flight.

The total time of flight is given by twice the time taken to reach the maximum height:

Total time of flight = 2*t

Let's calculate this.

First, we need to calculate V0y (the initial vertical velocity):

V0y = initial speed * sin(angle)

V0y = 1860 m/s * sin(67.4°)

Let's calculate V0y:

V0y = 1860 m/s * sin(67.4°) = 1782.79 m/s

Next, we calculate the time taken to reach the maximum height:

t = V0y / g

t = 1782.79 m/s / 9.8 m/s^2

Let's calculate t:

t = 1782.79 m/s / 9.8 m/s^2 = 181.86 s

Finally, we calculate the total time of flight:

Total time of flight = 2*t

Total time of flight = 2 * 181.86 s

Now, let's convert the time to minutes:

Total time of flight = 2 * 181.86 s * (1 min / 60 s)

Let's calculate the total time of flight:

Total time of flight = 2 * 181.86 s * (1 min / 60 s) = 6.062 min

Therefore, the total time of flight of the shell, neglecting air resistance, is approximately 6.062 minutes.