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) To determine the maximum height of the wall that can be cleared, we need to find the angle at which the projectile should be launched. The formula for the maximum height of a projectile launched at an angle is given by:
h_max = (v0^2 * sin^2(θ)) / (2 * g)
Where:
- h_max is the maximum height of the projectile
- v0 is the initial speed of the projectile
- θ is the launch angle
- g is the acceleration due to gravity
In this case, since you can launch at any angle, we need to find the launch angle θ that maximizes the value of h_max. To do this, we can take the derivative of the equation with respect to θ and set it equal to zero to find the optimum angle:
d(h_max)/dθ = (v0^2 * sin(2θ)) / g
Setting d(h_max)/dθ = 0, we have:
sin(2θ) = 0
This equation is satisfied when 2θ = 0 or 2θ = π, since sin(0) = 0 and sin(π) = 0. Therefore, we have two possible launch angles:
θ = 0 or θ = π/2
Since launching at θ = 0 would result in firing the projectile straight into the ground, the only feasible angle is θ = π/2, which corresponds to firing the projectile vertically.
So, the formula to maximize the height of the wall is:
h_max = (v0^2 * sin^2(π/2)) / (2 * g) = (v0^2) / (2 * g)
b) If the formula gives zero height for a certain value of x, it means that there is no launch angle that can clear the wall. This corresponds to the scenario where the moat is too wide (x) for the given initial speed (v0) and gravity (g) for any launch angle to result in the projectile reaching a height above zero. The projectile will not be able to clear the wall in this case.
c) When shooting to clear a wall of maximum height per the formula in part a, the missile is at the peak of its trajectory when it clears the wall. This occurs because the maximum height is achieved when the projectile is launched vertically upward (θ = π/2). At this launch angle, the projectile reaches its maximum height before descending back down. So, when the projectile clears the wall, it is descending from its peak height.