This boy is cliff-jumping beside a waterfall in Udzungwa National Park, Tanzania. For safety reasons, he might want to know how far away from the cliff he will land. We can calculate this by applying the equations for projectile motion to his descent.

The projectile motion equations can be obtained from the equation for basic 2D motion:

rf = ri + vit + 0.5at^2

where r is his position, v is velocity, a is acceleration, and t is time. Since the boy's motion in the horizontal direction is independent of his vertical motion, we can apply the above equation to his motion in the x and y directions, giving us two new equations.

xf = xi + vixt + 0.5axt^2

yf = yi + viyt + 0.5ayt^2

The picture at right approximates the boy's path over time. Assuming he leaped forward off the rock and not up into the air at all, his initial velocity in the y direction is 0, while his initial velocity in the x direction is some value vix. We designate the rock ledge where he started as some height yi and the water as yf. In the x-direction, the rock ledge xi is 0, and the distance from the cliff is xf. Because there is no force acting in the x direction to slow the boy or speed him up, acceleration in the x direction (ax) is 0. Gravity is acting to accelerate the boy in the y-direction, so ay = -g. Filling these values into our two equations results in two new equations:

xf = 0 + vixt + 0

0 = yi + 0 - 0.5gt^2

Solving the second equation for t gives us equations for the boy's final distance from the base of the cliff (xf) and the time the boy's fall will take (t). They are dependent on knowing the boy's initial velocity off the cliff (vix), the height of the cliff (yi), and the acceleration due to gravity (g).

xf = vixt

t = [yi / (0.5g)]^0.5