Question

A machine gun is mounted on the top of a tower room high. At
what angle should the gun be inclined to cover a maximum
range of firing on the ground below? bullet speed is 150m/s

Answer 1

I think it should be 45°

this is because this is the best angle for everything. If you hit a ball and want it to go very far with little power, 45° is the appropriate angle.

I hope it helps............

Answer 2

Explanation:

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Let h= 100m

Here, 

u = 150 ms-1

And, Taking g = 10 ms^{-2} (approximately) 

Let 'θ' be the angle at which the machine gun should fire in order to cover a maximum distance. 

Then , the horizontal component of velocity = 150 cos θ 

And, the vertical component of velocity = 150 sin θ 

If 'T' is the time of flight, then , 

Horizontal range, R = (150 cos θ ) x T . 

The gun is mounted at the top of a tower 100 meters high . 

Let us regard the positive direction of the position-axis as to be along the line from the top of tower in downward direction. 

For motion along vertical :

Initial Velocity = -150 sin θ; 

Distance covered = + 100 m 

And, acceleration = + 10 ms^{-2}

In time 'T' , the machine gun shot will reach maximum height and then reach the ground. 

Now, 

S = ut + 1/2 at^{2}

Therefore, +100 = ( -150 sin θ ) T + 1/2x10xT^{2}

Or, T^{2}- ( 30 sin θ )T - 20 = 0 

Therefore, T = - ( - 30 sin θ ) ± { ( - 30 sin θ)2 - 4 x 1 x (-20) }1/2/2 

= 30 sin θ ± (900 sin2 θ + 80 )1/2/2 

Or, T = 15 sinθ ± (225 sin2θ + 20)1/2

Now, range will be maximum, if time of flight is maximum . 

Therefore, choosing positive sign ,we have 

T = 15 sin θ + ( 225 sin2θ+ 20 )1/2

Hence, horizontal range covered, 

R = 150 cos θ {15 sin θ+ ( 225 sin2θ + 20 )}1/2

The horizontal range is maximum, when θ = 45°. 

But in the present case , the machine gun is mounted at height of 100 m  

Therefore, R will not be maximum for θ = 45°. It will be maximum for some value of θ close to 45

If we calculate values of R by setting θ = 43, 43.5, 44, 45, 46 and 47, the values of R come out to be 2347 m, 2347.7 m, 2348 m, 2346 m, 2341 m and 2334 m respectively. 

Thus R is maximum for value of θ some where between 43.5 and 44. 

Therefore , the mean value of θ = (43.5 + 44)/2 = 43.75. 

The gun should be inclined at 43.75 to cover a maximum range of firing on the ground below.