Question

When is a particle in projectile motion at its maximum height? What is true about its velocity when this occurs? Why?

Answer 1

In idealized projectile motion, a particle is at its maximum height when its instantaneous `y`-velocity is equal to zero.

How come?

Well, for projectile motion, after the particle is launched, the only factor affecting its motion in any way (ideally) is the downward gravitational force exerted by the Earth, which gives the object a constant downward acceleration of magnitude `9.81` `"m/s"^2`.

With that being said, if the particle is thrown upward, it has an initial `y`-velocity that is positive (taking positive direction to be upward), and this velocity is constantly being changed due to the gravitational acceleration.

Since the acceleration is directed opposite to the initial `y`-velocity, the velocity will decrease at a constant rate, and will intuitively become `0` at some point (and then negative afterward).

You might recall that the slope on a position vs. time graph at any point is the instantaneous velocity at that point. For a projectile thrown upward, its trajectory will resemble that of an inverse parabola like one shown here:

Notice that the slope of a projectile's path is positive until we reach the maximum height. What has happened is the initial velocity is constantly being decreased (i.e. the slope of the trajectory is decreasing) due to the negative acceleration, and at the maximum on the graph, the slope there (and thus the instantaneous velocity) is `0`, and therefore this is the maximum height, because after this, the motion (and velocity) is directed downward.