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 = 12.0 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

y2 = 10x,

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 = 12 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=12t, so that

t=x/12; substitute this into the y equation and get

y=1/2 gt^2 = 1/2 g(x/12)^2 = 0.04 x^2

now, we want to know when this curve intersects y^2=10x

y=Sqrt[10x]=0.04x^2

square both sides

10 x= 0.0016 x^4

6250=x^3

=> x=18.42m

when x=18.42, y=Sqrt[10*18.42]=13.57m,

so the impact occurs 13.57 m down, 18.42 m beyond the ledge

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