A truck loaded with cannonball watermelons stops suddenly to avoid running over

Question

A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washed-out bridge (see figure). The quick stop causes a number of melons to fly off the truck. One melon rolls over the edge with an initial speed vi = 14.5 m/s in the horizontal direction. A cross section of the bank has the shape of the bottom half of a parabola with its vertex at the edge of the road, and with the equation y2 = (18.0 m)x, where x and y are measured in meters. What are the x- and y-coordinates of the melon when it splatters on the bank?

Explanation / Answer

we need to write the equation of motion of the projectile;

let's call down the positive y direction to make life simple and consistent,

for the component of the projectile, we have that x = 14.5 t

and for the y value y=1/2 gt^2

now, we can make this an expression in x recalling that for this projectile, x=14.5t, so that

t=x/14.5; substitute this into the y equation and get

y=1/2 gt^2 = 1/2 g(x/14.5)^2 = 0.0233 x^2

now, we want to know when this curve intersects y^2=18x

y=Sqrt[18x]=0.0233x^2

square both sides

18 x= 0.00054289 x^4

33155.88=x^3 => x=32.125m

when x=32.125, y=Sqrt[18*32.125]=24.04m, so the impact occurs 24.04 m down, 32.124 m beyond the ledge

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