You are an artillery offcer attached to Gaius Julius Caesar's Xth Legion in the Gallic War
(1st C. BCE). You are responsible for a catapult, a device with a long spoonlike arm used
to lob stones or pumpkins over the walls of Gallic oppida. These fortresses are protected by
high walls rising from a deep, wide moat; you must fire your missiles from the far side of the
moat. You are expected to fire forward.
a) Caesar has asked you for an algebraic formula for the maximum height of wall you
can clear from across a moat of width x, if the initial speed of your projectile is v0,
the magnitude of the acceleration of gravity is g, and you can launch at any angle you
choose. What formula do you give him?* Assume aerodynamic forces are negligible
and disregard the height of the catapult itself.
b) For a certain value of x your formula gives zero height. Explain why-to what does
this correspond?
c)When you shoot to clear a wall of maximum height per the formula of part a, is your
missile ascending, descending, or at the peak of its trajectory when it clears the wall?
a) The algebraic formula for the maximum height of the wall that can be cleared from across a moat of width x, with an initial speed of v0 and the acceleration of gravity being g, is calculated using the principles of projectile motion.
To find the maximum height of the wall, you need to determine the angle at which to launch the projectile. Let θ represent the launch angle. The horizontal distance covered across the moat will be x, and the vertical distance (maximum height) reached will be denoted by h.
Using the equations of projectile motion, we can determine the maximum height:
1. The horizontal distance is given by x = (v0^2 * sin(2θ)) / g.
2. The vertical distance is given by h = (v0^2 * sin^2(θ)) / (2g).
By manipulating these equations, we can solve for h in terms of x, v0, and g:
h = (v0^2 * sin^2(θ)) / (2g) = (x * tan(θ)) - (g * x^2) / (2v0^2 * cos^2(θ)).
This is the algebraic formula for the maximum height of the wall that can be cleared.
b) If the formula gives a zero height for a certain value of x, it means that the projectile launched at any angle will not clear the wall. This corresponds to the scenario where the moat width is greater than the maximum horizontal distance the projectile can cover. In this case, no matter what angle the projectile is launched at, it will not be able to reach the top of the wall and hence have a zero height.
c) When you shoot to clear a wall of maximum height, as determined by the formula in part a, the missile is at the peak of its trajectory when it clears the wall. At the peak, the vertical velocity component becomes zero, and from there, the projectile starts descending due to the gravitational force. In order to clear the wall, the projectile needs to reach its maximum height before descending.