Ratio of the ranges of the bullets fired from a gun(a constant muzzle speed) at angle x, 2x and 4x is found in the ratio x:2:2 then the values of x will be(assume same muzzle speed of bullets)
2/√3
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give me the procedure of this answer
To solve this problem, we need to understand the concept of projectile motion and the range of a projectile.
The range of a projectile is the horizontal distance it travels from its launch point to its landing point. It depends on various factors, such as the initial velocity, angle of projection, and gravitational acceleration.
Let's assume that the constant muzzle speed of the bullets fired from the gun is 'v'. Since the muzzle speed is the same for all bullets fired, we can consider it as a constant.
Now, let's calculate the range of the bullet fired at angle 'x'. The horizontal component of the velocity will be v * cos(x), and the vertical component will be v * sin(x). The time of flight (the time taken by the bullet to reach the ground) can be calculated using the formula:
T = 2 * (v * sin(x)) / g,
where 'g' is the acceleration due to gravity.
The range (R1) of the bullet fired at angle 'x' can be calculated by multiplying the horizontal component of the velocity (v * cos(x)) by the time of flight:
R1 = (v * cos(x)) * T.
Similarly, we can calculate the ranges (R2 and R3) for the bullets fired at angles 2x and 4x, respectively.
R2 = (v * cos(2x)) * T.
R3 = (v * cos(4x)) * T.
We are given that the ratio of the ranges is x:2:2. So, we can write:
R2 / R1 = x / 2.
R3 / R1 = 2 / 2.
Substituting the range equations, we get:
[(v * cos(2x)) * T] / [(v * cos(x)) * T] = x / 2.
[(v * cos(4x)) * T] / [(v * cos(x)) * T] = 2 / 2.
Canceling out the common terms, we have:
cos(2x) / cos(x) = x / 2.
cos(4x) / cos(x) = 1.
Now, solving the equations will give us the values of 'x':
cos(2x) / cos(x) = x / 2.
cos(2x) = (x / 2) * cos(x).
cos(2x) = (1/2) * x * cos(x).
Let's plot the graphs of y = (x / 2) * cos(x) and y = cos(2x) to find their points of intersection, which will give us the possible values of 'x'.
To do this, we can use graphing software or an online graphing tool. Plotting the graphs will help us determine the values of 'x' where they intersect.
Once the intersection points are obtained, those will be the possible values for 'x' that satisfy the given ratio of ranges.
Please note that due to the nature of the equation, it might not be possible to find exact values for 'x.' In such cases, you can approximate the values using numerical methods or calculators that can solve trigonometric equations.