A trained dolphin leaps from a pool of water. With the correct initial speed and direction the dolphin can catch a falling ball in midair.
Equations to solve this are below the applet.
The equation of the ball's motion is easy because it is simply vertical:
yball = (1/2)(-9.8 m/s2)t2 + y0
The equation of the dolphin's motion, however, is a bit more tricky. We need to break it up to it's horizontal and vertical components.
Vertically the initial velocity is v0y = v0sin(θ)
Therefore the dolphin's vertial position is:
ydolphin = (1/2)(-9.8 m/s2)t2 + v0sin(θ)t
The horizontal position is much more simple because there is no acceleration in the "x-direction". The velocity in the x direction (not initial velocity because the velocity remains the same in the x direction) is:
vx = v0cos(θ)
xdolphin = v0cos(θ)t
The final xdolphin is the ball's horizontal position, so it is easy enough to solve for the time for the dolphin to catch the ball:
t = xfinal/[v0cos(θ)]
Because the dolphin will catch the ball (we hope!), we can set the final vertical distance of the dolphin equal to that of the ball and solve:
(1/2)(-9.8 m/s2)t2 + v0sin(θ)t = (1/2)(-9.8 m/s2)t2 + y0
And this just becomes:
v0sin(θ)t = y0
where on the left side of the equation it refers to the dolphin's initial speed and angle at which it leave the water, and the right side refers to the ball's initial height.
From there, solving is somewhat easy:
v0sin(θ)xfinal/[v0cos(θ)] = y0
This simplifies to tan(θ) = y0/xfinal
Or another way to say it is that the initial speed of the dolphin doesn't matter that much (as long as it has enough speed to catch the ball before it hits the water).
So the angle is just θ = tan-1(y0/xfinal)
If we use the default settings in the applet:
θ = tan-1(4.1/5.5) = 36.7°.