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I don’t understand why tangential acceleration changes in value in a parabolic movement with constant acceleration (gravitational acceleration). Since acceleration is constant, tangential and centripetal acceleration are either both constant or one increases and other decreases in value. I was told that they both change in such movements but I don’t understand why since Tangential Acceleration is the derivative of the velocity value. Isn’t velocity supposed to change at the same rate in this case? And, since it changes, in which point of the trajectory is tangential acceleration at its maximum value?

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In free fall without any horizontal motion, gravity changes the tangential velocity (as the path is vertical). The same applies to all situations unless the path is perpendicular to gravity.

This is illustrated in this post where tangential acceleration is

$$ \dot{v} = \vec{e} \cdot \vec{g} $$ where $\vec{e}$ is the tangent vector.

For example:

Path

The velocity vector is tangent to the path

$$ \vec{v} = v\,\vec{e} $$

where $v$ is the speed

The acceleration (gravity) is decomposed into

$$ \vec{g} = \dot{v} \vec{e} + \frac{v^2}{r} \vec{n} $$

with $r$ the radius of curvature of the path.

with $$ \begin{aligned} \dot{v} & = \vec{e}\cdot \vec{g} \\ \frac{v^2}{r} & = \vec{n} \cdot \vec{g} \end{aligned} $$

Only if $\vec{e}\cdot\vec{g} = 0$ the tangential acceleration is zero (since the motion is perpendicular to gravity).

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  • $\begingroup$ I am talking about vertical and horizontal motion - parabolic trajectory $\endgroup$
    – opbravo
    Oct 22, 2018 at 17:18
  • $\begingroup$ As I said, as long as there is a component of gravity along the path, the tangential acceleration is going to change. I edited the post with lots more details. $\endgroup$ Oct 22, 2018 at 18:30
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Since the acceleration g is always vertical and the direction of a parabolic trajectory is constantly changing, the ratio between the tangential and the centripetal (normal) acceleration would be changing as well.

At the top of a trajectory (the maximum of an inverted parabola), g is normal to the trajectory, so the tangential acceleration at this point is zero.

As we move down along the inverted parabola, the tangential component of g is increasing, while the normal component is decreasing. The tangential component will be approaching g, its theoretical maximum, as the trajectory gets closer to vertical.

enter image description here

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