Displacement Diagram


A ball is kicked directly upward. After the ball reaches its highest altitude, it falls back down to its initial position because of gravity. The altitude function of the ball is s = 30t - 5t2. How high can the ball go up? The for displacement and velocity units are m and m/s, respectively.

Since the ball starts and ends at the same vertical location, its displacement function is continuous and differentiable. This satisfies the prerequisite of Rolle's Theorem, and thus a point exists so that the derivative with respect to time, t, is 0.

     ds/dt = d(30t -5t2)dt

             = d(30t)/dt - d(5t2)/dt

             = 30 - 10t

As mentioned this must equal zero,

ds/dt = 30 - 10t = 0

Since the derivative of displacement with respect to time is velocity, the above equation can be rewritten as

     ds/dt = v = 30 - 10t = 0

Therefore, t = 3

Substituting t = 3 into the displacement function
     s = 30t - 5t2

        = 30(3) - 5(3)2

        = 45 m

Thus, the maximum altitude of the ball will be 45 m.