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 v_i = 10.0 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 initial location of the projected watermelon and with the equation y^2 = 12x, 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

let's call down the positive y direction
for the component of the projectile, we have that x = 10 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=10t, so that

t=x/10; substitute this into the y equation and get

y=1/2 gt^2 = 1/2 g(x/10)^2 = 0.049 x^2

now, we want to know when this curve intersects y^2=12x

y=Sqrt[12x]=0.049x^2

square both sides

12 x= 0.002401 x^4

4997=x^3

=> x=17.10m

when x=17.10,

y=Sqrt[12*17.10]=14.32m, so the impact occurs 14.32 m down, 17.10 m beyond the ledge

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