the range of a rifle bullet is 100m, when thither is the angle of projection if the bullet is fired withe the same angle from a car traveling at 36km/h towards the target, show that the range will increase by 100/7 squire root of tan thither
To show that the range will increase by 100/7 times the square root of tan(theta), we need to analyze the situation using projectile motion equations.
Let's start by considering the range of the bullet when fired from a stationary position. The range, R, of a projectile fired at an angle, theta, from the ground can be calculated using the formula:
R = (v^2 * sin(2*theta)) / g
Where:
- v is the initial velocity of the projectile (bullet)
- g is the acceleration due to gravity (9.8 m/s^2)
In this case, we are given the range R as 100m. So,
100 = (v^2 * sin(2*theta)) / g
Next, let's consider the scenario where the bullet is fired at the same angle, theta, from a car traveling at 36 km/h (10 m/s) towards the target. The bullet is already moving with the velocity of the car in the forward direction. So, the initial velocity, v, can be calculated as:
v = v_car + v_bullet
v = 10 + v_bullet
Now, let's determine the range, R', of the bullet fired from the moving car. The range R' can also be calculated using the same formula as before. However, we need to consider the relative motion between the car and the ground. The bullet and car are both moving in the same direction, so the effective velocity of the bullet relative to the ground is equal to the difference in velocities:
v_eff = v - v_car
v_eff = v_bullet
Therefore, we can substitute v_bullet for v and calculate R':
R' = (v_bullet^2 * sin(2*theta)) / g
Now, we can compare the ranges R and R' to see how they are related:
R' / R = [(v_bullet^2 * sin(2*theta)) / g] / [(v^2 * sin(2*theta)) / g]
= (v_bullet^2 / v^2)
Substituting v = 10 + v_bullet:
R' / R = (v_bullet^2 / (10 + v_bullet)^2)
Next, we can simplify the equation by dividing both the numerator and denominator by v_bullet^2:
R' / R = (1 / (1 + (v_bullet/10))^2)
Now, substituting R = 100:
R' / 100 = (1 / (1 + (v_bullet/10))^2)
We need to show that the range increases by 100/7 times the square root of tan(theta). Let's simplify further:
R' / 100 = (1 / (1 + (v_bullet/10))^2)
= 1 / ((1 + (v_bullet/10))^2)
= 1 / (1 + 2*(v_bullet/10) + (v_bullet/10)^2)
= 1 / (1 + 2*(v_bullet/10) + (v_bullet^2/100))
Now, let's compare this with the expression we need to show:
100/7 * sqrt(tan(theta))
To establish the connection, we need to equate the expressions on both sides. So,
1 / (1 + 2*(v_bullet/10) + (v_bullet^2/100)) = 100/7 * sqrt(tan(theta))
To solve this equation, we can cross-multiply and rearrange the terms:
7 * sqrt(tan(theta)) = 1 + 2*(v_bullet/10) + (v_bullet^2/100)
7 * sqrt(tan(theta)) - 1 = 2*(v_bullet/10) + (v_bullet^2/100)
700 * sqrt(tan(theta)) - 100 = 20*v_bullet + v_bullet^2
Now, we have an equation relating the angle of projection, theta, with the velocity of the bullet, v_bullet. By solving this quadratic equation for v_bullet, we can get the velocity required to increase the range by the specified factor.
Please note that this is the derivation of the relationship between the range, angle of projection, and velocity in the given scenario. The actual calculation of the increased range requires knowing the value of theta and solving the equation mentioned above.