Ball fired from cannon homework

In summary, the conversation is about finding the optimal angle (θ) for a cannon to fire a ball at a maximum horizontal distance (d) when the cannon is aimed up along an incline of length L. The person speaking has attempted to find the solution but has not been successful and is looking for help before the homework deadline.
  • #1
zippyjibby
1
0

Homework Statement


A cannon, when aimed vertically, is observed to fire a ball to a maximum height of L. Another ball is then fired with the same speed, but with the cannon aimed up along an incline of length L, inclined at an angle θ. What should θ be so that the ball travels the largest horizontal distance d by the time it returns to the height of the top of the plane?

2. The attempt at a solution
My attempt is a bit messy, as you can see below, but I tried to find an equation to describe the initial velocity at the base of the ramp, and an equation to describe the velocity as it leaves the ramp. Then I plugged that into basic projectile motion equations, and got d in terms of constants and θ. I then took the deriviative to find the optimal angle, but I could not figure out the equation. The homework is due Friday (9/18/2015) at 5pm, and I have not been able to figure it out.
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  • #2
zippyjibby said:

Homework Statement


A cannon, when aimed vertically, is observed to fire a ball to a maximum height of L. Another ball is then fired with the same speed, but with the cannon aimed up along an incline of length L, inclined at an angle θ. What should θ be so that the ball travels the largest horizontal distance d by the time it returns to the height of the top of the plane?

2. The attempt at a solution
My attempt is a bit messy, as you can see below, but I tried to find an equation to describe the initial velocity at the base of the ramp, and an equation to describe the velocity as it leaves the ramp. Then I plugged that into basic projectile motion equations, and got d in terms of constants and θ. I then took the deriviative to find the optimal angle, but I could not figure out the equation. The homework is due Friday (9/18/2015) at 5pm, and I have not been able to figure it out. View attachment 88781View attachment 88782View attachment 88783
View attachment 88784

I, for one, seldom attached photos of rough work, and never those that are sideways. I will look at it if you type it out.
 

1. How do you calculate the velocity of a ball fired from a cannon?

The velocity of a ball fired from a cannon can be calculated using the formula v = √(2gh), where v is the velocity, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the cannon relative to the ground. This assumes that there is no air resistance and the cannon is fired horizontally.

2. What is the maximum height reached by a ball fired from a cannon?

The maximum height reached by a ball fired from a cannon can be calculated using the formula h = (v^2)/(2g), where h is the maximum height, v is the initial velocity, and g is the acceleration due to gravity. This assumes that there is no air resistance and the cannon is fired at an angle of 45 degrees.

3. How do you calculate the time of flight for a ball fired from a cannon?

The time of flight for a ball fired from a cannon can be calculated using the formula t = 2v/g, where t is the time of flight, v is the initial velocity, and g is the acceleration due to gravity. This assumes that there is no air resistance and the cannon is fired horizontally.

4. What factors affect the trajectory of a ball fired from a cannon?

The trajectory of a ball fired from a cannon is affected by several factors, including the initial velocity, angle of elevation, air resistance, and the mass and shape of the ball. Other factors such as wind and temperature can also have an impact on the trajectory.

5. How does air resistance affect the trajectory of a ball fired from a cannon?

Air resistance can significantly affect the trajectory of a ball fired from a cannon. As the ball moves through the air, it experiences a force in the opposite direction of its motion, which can cause it to slow down and deviate from its intended path. This can be accounted for by using more complex equations or by conducting experiments to measure the effects of air resistance.

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