You are a pirate working for Dread Pirate
Roberts. You are in charge of a cannon that
exerts a force 20000 N on a cannon ball while
the ball is in the barrel of the cannon. The
length of the cannon barrel is 1.67 m and
the cannon is aimed at a 47◦
angle from the
ground.
The acceleration of gravity is 9.8 m/s^2
If Dread Pirate Roberts tells you he wants
the ball to leave the cannon with speed v0 =
72 m/s, what mass cannon ball must you use?
Answer in units of kg.
Assuming the Dread Pirate Roberts never
misses, how far from the end of the cannon is
the ship that you are trying to hit (Neglect
dimensions of cannon)?
Answer in units of m
let M = mass of cannon ball ... and r = range
(work done by cannon) - (work against gravity) = energy of cannon ball
(20 kN * 1.67 m) - [M * g * 1.67 m * sin(47º)] = 1/2 * M * (72 m/s)^2
solve for M
r = (72 m/s)^2 * sin(2 * 47º) / g
To find the mass of the cannonball, we can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a). In this case, the force exerted on the cannonball is 20,000 N, and the acceleration due to gravity is 9.8 m/s^2.
We can rearrange the equation to solve for mass: m = F / a
Substituting the values, we have m = 20,000 N / 9.8 m/s^2
Calculating this gives us a mass of approximately 2040.8 kg for the cannonball.
To determine the distance from the end of the cannon to the ship, we need to use some trigonometry. We know that the cannon is aimed at a 47° angle from the ground, and we want to find the horizontal distance (x).
We can use the following trigonometric equation: x = v0 * t * cos(theta), where v0 is the initial velocity, t is the time of flight, and theta is the angle of elevation.
The initial velocity is given as 72 m/s, and the angle of elevation is 47°.
To find the time of flight, we can use the vertical component of the motion. The vertical distance (y) traveled can be calculated using a kinematic equation: y = v0 * t * sin(theta) - (1/2) * g * t^2, where g is the acceleration due to gravity.
Since the ball is launched at an angle and lands at the same height, the vertical displacement (y) will be zero. Therefore, we can set the equation to 0 and solve for t.
0 = (72 m/s) * t * sin(47°) - (1/2) * (9.8 m/s^2) * t^2
Rearranging and factoring, we get t * (36 * sin(47°) - 4.9 * t) = 0
Solving this quadratic equation, we find two possible values for t: t = 0 and t = (36 * sin(47°)) / 4.9
Since time cannot be zero in this case, we can discard the t = 0 solution.
Substituting t into the equation for x, we have x = (72 m/s) * ((36 * sin(47°)) / 4.9) * cos(47°)
Calculating this gives us an approximate distance of 632.39 m from the end of the cannon to the ship that you are trying to hit.
So, the mass of the cannonball is 2040.8 kg, and the ship is approximately 632.39 m away from the end of the cannon.
To find the mass of the cannonball, we can use the equation of motion for the projectile. The force exerted on the cannonball is equal to the product of its mass and acceleration:
Force = mass * acceleration
The force exerted on the cannonball is 20,000 N. The acceleration of gravity is 9.8 m/s².
Using this equation, we can rearrange it to solve for the mass of the cannonball:
mass = Force / acceleration
mass = 20,000 N / 9.8 m/s²
mass ≈ 2,040.82 kg
Therefore, the mass of the cannonball is approximately 2,040.82 kg.
To find the distance from the end of the cannon to the ship, we can use the equations of motion for projectile motion. The vertical and horizontal motions are independent of each other. In this case, we are interested in the horizontal distance traveled by the cannonball.
The formula for the horizontal distance (range) of a projectile is given by:
Range = (initial velocity)² * sin(2 * angle) / gravity
The initial velocity is 72 m/s, the angle is 47º, and the acceleration due to gravity is 9.8 m/s².
Using this formula, we can calculate the range:
Range = (72 m/s)² * sin(2 * 47º) / 9.8 m/s²
Range ≈ 503.29 m
Therefore, the distance from the end of the cannon to the ship is approximately 503.29 m.