Cannonball Projectile Motion

In summary, the maximum horizontal range of a cannon with a launch speed of 500m/s and aimed at a target 2.8km away can be represented by the function v^2sin2(theta)/g. To determine the launch angle needed to hit the target, the values can be plugged into the formula. The velocity of the cannonball when it hits its target can be determined by finding the final velocity. The speed of the cannonball's shadow along the ground can be determined by finding its velocity, assuming the sun is shining vertically.
  • #1
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Homework Statement



A cannon has a launch speed of 500m/s and is aimed at a target 2.8km away.
a. Write an equation for the maximum horizontal range of the cannon as a function of the launch angle theta.
b. At what angle must the cannon be launched in order to hit the target?
c. As viewed from above, how quickly does the cannonball's shadow move along the ground?
d. How fast is the cannonball moving when it hits its target?


Homework Equations



D = vit+1/2at^2
velocity formulas

Range of a projectile

The Attempt at a Solution



a. I used the range of a projectile motion and got v^2sin2(theta)/g but I don't know how to make it a function of the launch angle theta. Am I overreading?
b. I am assuming you just plug in the values from the first formula you get.
c. Just find the acceleration?
d. Just find the vf?
 
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  • #2
a. I used the range of a projectile motion and got v^2sin2(theta)/g but I don't know how to make it a function of the launch angle theta. Am I overreading?
d= v^2sin2(theta)/g is a formula for the range as function of the launch angle (and constants). I used d= with d as range to make a formula out of your expression.
b. I am assuming you just plug in the values from the first formula you get.
Right.
c. Just find the acceleration?
Why acceleration?
d. Just find the vf?
There is an easier way, but that works as well.
 
  • #3
Because it asks how quickly does the cannonball's shadow move along the ground so I am assuming that means acceleration? Because its velocity is constantly changing so it can't be that
 
  • #4
Because it asks how quickly does the cannonball's shadow move along the ground so I am assuming that means acceleration?
No, that certainly means velocity. The velocity of the shadow is not changing here if the sun is shining vertically.
 
  • #5



a. Yes, you are correct in using the range formula for projectile motion. To make it a function of the launch angle theta, you can simply replace the angle in the formula with theta. So your equation would be: R = (v^2sin2(theta))/g, where R is the range and v is the launch speed.

b. To find the angle at which the cannon must be launched to hit the target, you can rearrange the formula to solve for theta. So your equation would be: theta = (1/2)arcsin(gR/v^2). Plugging in the values of g, R, and v, you can calculate the angle theta.

c. To find how quickly the cannonball's shadow moves along the ground, you can use the velocity formula for projectile motion, which is v = v0 + at. Since the cannonball's initial velocity is 500m/s and the acceleration due to gravity is 9.8m/s^2, the velocity of the cannonball at any given time will be 500m/s - 9.8m/s^2t. The shadow will move at the same velocity as the cannonball, so the speed of the shadow can be calculated using the same formula.

d. To find the speed of the cannonball when it hits its target, you can use the velocity formula again. This time, you will be solving for the final velocity, vf. So your equation would be: vf = v0 + at. Plugging in the values of v0 (500m/s) and a (9.8m/s^2), you can calculate the final velocity vf.
 

What is cannonball projectile motion?

Cannonball projectile motion refers to the motion of a spherical object, such as a cannonball, that is launched into the air at an angle with a certain initial velocity.

What factors affect cannonball projectile motion?

The factors that affect cannonball projectile motion include the initial velocity, angle of launch, air resistance, and gravitational force.

How is the trajectory of a cannonball calculated?

The trajectory of a cannonball can be calculated using the equations of motion, taking into account the initial velocity, angle of launch, and acceleration due to gravity.

What is the maximum height reached by a cannonball?

The maximum height reached by a cannonball is determined by the initial velocity and angle of launch. It can be calculated using the equation h = (v^2 * sin^2θ)/(2g), where h is the maximum height, v is the initial velocity, θ is the angle of launch, and g is the acceleration due to gravity.

How does air resistance affect cannonball projectile motion?

Air resistance can affect cannonball projectile motion by slowing down the cannonball and altering its trajectory. The greater the air resistance, the shorter the distance the cannonball travels. This can be taken into account by using more complex equations or conducting experiments to measure the air resistance and adjusting the initial velocity accordingly.

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