A shell is fired from the ground level with muzzle speed of 750 ft/s at an angle of . An
enemy gun 20,000 ft away fires a shot 2 seconds later and the shell collide 50 ft above the
ground at the same speed. What are the muzzle speed v and angle of elevation alpha of the second
gun?
To solve this problem, we can assume that the shell travels in a parabolic trajectory and use the equations of motion for projectile motion.
Let's first find the time it takes for the first shell to reach the point of collision. The horizontal distance traveled by the shell is 20,000 ft, and the initial horizontal velocity is the muzzle speed (750 ft/s) multiplied by the cosine of the angle of elevation.
Horizontal distance = initial horizontal velocity * time
20000 = 750 * cos(alpha) * t1
Simplifying, we have:
t1 = 20000 / (750 * cos(alpha))
Next, let's find the time it takes for the enemy shell to reach the point of collision. The horizontal distance traveled by the enemy shell is also 20,000 ft, and the initial horizontal velocity is assumed to be v multiplied by the cosine of the angle of elevation.
Horizontal distance = initial horizontal velocity * time
20000 = v * cos(alpha) * t2
Simplifying, we have:
t2 = 20000 / (v * cos(alpha))
Since the enemy shell is fired 2 seconds later, we can write:
t2 = t1 - 2
Combining the equations for t1 and t2, we have:
20000 / (v * cos(alpha)) = 20000 / (750 * cos(alpha)) - 2
Dividing both sides by 20000, we get:
1 / (v * cos(alpha)) = 1 / (750 * cos(alpha)) - 1/10000
Multiplying both sides by (v * cos(alpha)), we have:
1 = v / 750 - (v / 10000) * cos(alpha)
Simplifying further, we get:
(v / 750) * cos(alpha) + (v / 10000) * cos(alpha) = 1
Multiplying through by 750 * 10000, we have:
10000 * v * cos(alpha) + 750 * v * cos(alpha) = 750 * 10000
Combining like terms, we get:
10750 * v * cos(alpha) = 750 * 10000
Dividing both sides by 10750 * cos(alpha), we have:
v = (750 * 10000) / (10750 * cos(alpha))
Finally, we get the equation for muzzle speed v:
v = 70 / (107 * cos(alpha))
Now we can use this equation to calculate the muzzle speed v and angle of elevation alpha.