How to describe characteristics a parabola?

[decadyne]

Help me understand the components of a parabola as created by a physical object in normal gravity.

The object has some forward velocity, some upward velocity, and at some point some downward velocity. Is this correct?

Is it safe to say, therefore, that at some point the parabola's vertical-axis velocity is zero? This is at the apex, correct?

Please tell me more about how to describe it, and if my assumptions are correct. Are there any other places where its characteristics can be predicted?

 

[Xezlec]

The parabola you seem to be discussing is the trajectory of a mass with some initial velocity in an environment with uniform gravity and no friction. The properties you have listed are correct. But when you say "the parabola's velocity", what you mean to say is either "the object's velocity" or "the parabola's derivative". A derivative is the mathematical property of a geometric shape that corresponds to the concept of velocity of the object in this case.

All of the characteristics of the parabolic trajectory can be predicted. Are there any in particular that you are curious about? In this case, the forward velocity is always the same (since there is no friction, and since gravity has no component in that direction and therefore no effect in that direction). The vertical velocity at every point in time can be computed using the definitions of force (F=dp/dt) and momentum (p=mv) and a little simple calculus, bearing in mind that "uniform gravity" means the gravitational force is always downward and always the same number. Specifically,

F = dp/dt = d(mv)/dt = m dv/dt.

Then, integrating both sides,

t F = m (v + v0).

So,

v = v0 + (t F / m),

where v is the upward velocity at time t, v0 is the upward velocity at time t=0, F is the force of gravity (which should be a negative number because it is downward), and m is the mass of the object. Since the force due to gravity on Earth is g m, where g=-9.8 m/s/s, we can be even more specific:

v = v0 + (t g m / m) = v0 + t g = v0 - 9.8 t,

Where v and v0 are in m/s and t is in seconds.

Does that about clear it up?