Suppose you toss a ball upward into the air and then catch it at the same point where you tossed it from. Must the ball's velocity have been zero at some point? {{CMD_39}} Let {{MATH_40}} be the position of the ball (on the vertical line) at time {{MATH_41}}. Our time interval in question will be {{MATH_42}} We may assume that {{MATH_43}} is continuous on {{MATH_44}} and differentiable on {{MATH_45}}. Since {{MATH_46}}, we may now apply Rolle's Theorem and conclude that at some time {{MATH_47}} in the given time interval, {{MATH_48}}. Hence the velocity must be zero at some point. {{CMD_49}} {{CMD_50}}