The cannon on a pirate ships shoots cannon balls with a speed of 350m/s (the muzzle velocity). The cannon can be adjusted to shoot at any elevation above the horizontal. What is the greatest kinetic energy the cannon ball can have when it hits its target in J ? Ignore air resistance and assume the target is at the same height above the ground (or water) as the cannon. mass of cannon ball is 5kg
To find the greatest kinetic energy of the cannonball when it hits its target, we need to determine the angle of elevation that maximizes the range (horizontal distance) of the cannonball.
The range of a projectile can be calculated using the equation:
R = (v² * sin(2θ)) / g
Where:
R = Range
v = Muzzle velocity (350 m/s)
θ = Angle of elevation
g = Acceleration due to gravity (9.8 m/s²)
Since we want to maximize the range, we need to find the angle θ that produces the maximum value for sin(2θ).
To do this, we take the derivative of sin(2θ) with respect to θ and set it equal to zero to find the critical points. Then we determine whether these critical points are maximum or minimum points.
d(sin(2θ))/dθ = 2cos(2θ) = 0
cos(2θ) = 0
2θ = π/2, 3π/2, 5π/2, ...
To maximize the range, we consider the first positive value for θ since negative angles will give the same range. Thus,
2θ = π/2
θ = π/4
Now, we can calculate the range R using:
R = (v² * sin(2θ)) / g
R = (350² * sin(2 * π/4)) / 9.8
R ≈ 62560 m
Finally, we can calculate the kinetic energy of the cannonball using the formula:
KE = (1/2) * m * v²
KE = (1/2) * 5 * (350)²
KE ≈ 306250 J
Therefore, the greatest kinetic energy the cannonball can have when it hits its target is approximately 306,250 Joules.