Calculating Range of a Cannon at an Angle

[jli10]

Homework Statement

A cannon fires projectiles on a flat range at fixed speed but with variable angle. The maximum range of the cannon is L. What is the range of the cannon when it fires at an angle $\frac{\pi}{6}$ above the horizontal? Ignore air resistance.

Homework Equations

Four kinematic equations.

The Attempt at a Solution

I honestly don't know where to start. What exactly does the question mean by the range of the cannon? Is that the horizontal distance the projectile goes?

 

[ActionFrank]

Yes the range will be the horizontal distance. Since projectiles move horizontally at a constant velocity, it is simply calculated by multiplying the x-component of the velocity by the time of flight.

x=v(x)*t
where v(x) means the x-component of the velocity. v(x) = v*cos(theta)
x = the range

The trickier part is calculating the time of flight. This is done by finding the y-component of your initial velocity, then recognizing that the final velocity at landing will be the opposite of this value. Knowing that the projectile will accelerate at -9.8 m/s^2 you can then solve for the time of flight and substitute it back into the first equation.

v(y-final) = v(y-initial) + a*t
v(y-initial) = v*sin(theta)
v(y-final) = -v(y-initial)

Hope that helps.

 

[jedishrfu]

range is how far away the target is. Given the maximum range is L then they are looking for a percentage of L when the angle is pi/6.

show some work and people will help.

 

[Nugatory]

Yes, the "range" is the horizontal distance the projectile travels.

What angle will produce the maximum range? What is the speed of the projectile if the maximum range is L?

 

[asteriatic]

I can provide a response to the following content by first clarifying the question and then providing a solution using the given information and relevant equations.

The question is asking for the horizontal distance that the projectile fired from the cannon will travel when it is fired at an angle of π/6 (30 degrees) above the horizontal. The given information is that the cannon fires projectiles with a fixed speed and has a maximum range of L. We can assume that the cannon is fired from ground level and that air resistance is negligible.

To solve this problem, we can use the kinematic equation for range, which is given by R = (v^2/g) * sin(2θ), where v is the initial velocity, g is the acceleration due to gravity, and θ is the firing angle. We are given the maximum range, L, which means that we can set this value equal to R and solve for v. This will give us the initial velocity of the projectile.

Once we have the initial velocity, we can plug it into the range equation along with the firing angle of π/6 and solve for R. This will give us the range of the cannon when it is fired at an angle of π/6 above the horizontal.

It is important to note that this solution assumes ideal conditions and does not take into account factors such as air resistance or changes in elevation. To get a more accurate range, these factors would need to be considered and additional equations or simulations may be necessary.