A target is 10km from a gun and the gun's muzzle speed is 400m/s. What angle of projection will ensure a hit?
To solve this problem, we can use the equations of motion for projectile motion. The horizontal distance covered by the projectile can be given by the equation:
range = (initial velocity x time) x cos(angle),
where range is the horizontal distance covered, initial velocity is the muzzle speed of the gun (400m/s), time is the time of flight, and angle is the angle of projection.
In this case, the range is given as 10km, which can be converted to meters by multiplying by 1000:
10km = 10,000m.
Now, we can rewrite the equation as:
10,000m = (400m/s x time) x cos(angle).
We also know that the vertical distance covered by the projectile can be given by the equation:
vertical distance = (initial velocity x time) x sin(angle) - (0.5 x gravity x time^2),
where gravity is the acceleration due to gravity (approximately 9.8m/s^2).
In this case, we want to ensure that the projectile hits the target, which means the vertical distance covered should be zero. Therefore, we have:
0 = (400m/s x time) x sin(angle) - (0.5 x 9.8m/s^2 x time^2).
Now, we can solve this system of equations to find the angle of projection that will ensure a hit.
First, we can rearrange the second equation to isolate time:
0.5 x 9.8m/s^2 x time^2 = (400m/s x time) x sin(angle).
Simplifying further:
4.9m/s^2 x time = 400m/s x sin(angle).
Now, we can substitute the value of time from this equation into the first equation:
10,000m = (400m/s x (4.9m/s^2 x time) / sin(angle)) x cos(angle).
Now, we can solve this equation to find the angle of projection that will ensure a hit. However, this is a complex equation to solve analytically. Therefore, we can solve it numerically or use approximation methods such as trial and error or Newton's method to find an approximate solution.
Using trial and error or numerical methods, the angle of projection that will ensure a hit is approximately 1.41 degrees (or 0.025 radians) with an initial velocity of 400m/s.