When we throw an object up in the air its velocity at the peak?

An alternative approach to this is to consider a theorem in analysis known as the intermediate value theorem: this states, in its simplest form, that, given a function $f$ which is continuous on some interval $[a, b]\subset \mathbb{R}$, then, if $f(a) \le f(b)$, for any $y \in [f(a), f(b)]$ there is an $x \in [a, b]$ such that $f(x) = y$, and equivalently if $f(a) \ge f(b)$.

Well, upward velocity in this case is continuous as a function of time, and at the start of the trajectory it is positive and at the end it is negative: therefore, by the intermediate value theorem there must be some point in the trajectory where it is zero.

This may seem like an absurdly mathematical way of looking at the problem: physical intuition can tell you the same thing. But actually I think it's useful to try and apply analysis in these simple cases because it has the great property that it's made of theorems with clear assumptions ('velocity is a continuous function if time’) which then spit out unambiguous answers: this can be extremely useful in cases where physical intuition may be less useful, or unavailable.

QUESTION #90

Asked by: Robin Pleshek First, let's assume you are not taking air resistance into consideration. This is a reasonable assumption because it's obvious that a parachute's velocity just before it hits depends on factors other than just gravity or drop height. Adjusting its size can influence its speed at impact. Without air friction, an object WILL regain its original upward velocity just before it hits the ground on the way down. The initial velocity on the ground represents KINETIC ENERGY, and the initial height of 0 represents NO POTENTIAL ENERGY. As the object rises, its kinetic energy is gradually converted into potential energy until at its highest point all the original kinetic energy (now zero) has been converted into potential energy. At any point in its travel, TOTAL kinetic and potential energy remains constant. The process is reversed when the object falls. Starting at maximum potential energy (determined by its maximum height) and zero kinetic energy (not moving), the potential energy gradually gets converted back into kinetic energy until just before it hits the ground. Then, potential energy is zero and it has all the kinetic energy it had at the start. So the CONSERVATION OF ENERGY is why the two velocities are the same.

Answered by: Paul Walorski, B.A. Physics, Part-time Physics Instructor