KAHIBARO
Table of Contents
Motion Along the Vertical Direction
Vertical motion is motion up and down, along a vertical line. Near the surface of Earth, this kind of motion is especially important because gravity acts mainly downward and produces a nearly constant acceleration. This makes vertical motion a direct application of constant acceleration.
In vertical motion, the object may move upward, downward, or first upward and then downward. Even though the direction of motion can change, the acceleration due to gravity remains directed downward throughout the motion, as long as air resistance is neglected.
Choosing a Sign Convention
To solve vertical motion problems correctly, the most important step is choosing a coordinate direction. A common choice is to take upward as positive. Then downward is negative.
With that choice,
$$
a = -g
$$
where $g \approx 9.8\ \text{m/s}^2$.
If instead you choose downward as positive, then gravity becomes positive:
$$
a = +g
$$
Both choices work, but you must stay consistent from start to finish.
In vertical motion near Earth's surface, gravity always points downward. The sign of $g$ in equations depends on your chosen coordinate system, not on the motion of the object.
Using the Constant Acceleration Equations
Since vertical motion under gravity has constant acceleration, the usual kinematic equations apply directly. If upward is positive, replace $a$ by $-g$:
$$
v = v_0 - gt
$$
$$
y = y_0 + v_0 t - \frac{1}{2}gt^2
$$
$$
v^2 = v_0^2 - 2g(y - y_0)
$$
Here, $y$ is the vertical position, $y_0$ is the initial vertical position, $v$ is the vertical velocity, and $v_0$ is the initial vertical velocity.
These equations describe many situations, including throwing a ball upward, dropping an object, or launching an object downward.
For upward positive coordinates, the standard vertical motion equations are
$$
v = v_0 - gt
$$
$$
y = y_0 + v_0 t - \frac{1}{2}gt^2
$$
$$
v^2 = v_0^2 - 2g(y - y_0)
$$
Use them only when the acceleration is constant and air resistance is neglected.
Upward Motion
Suppose an object is thrown straight upward. Its initial velocity is positive if upward is chosen as positive. As the object rises, gravity slows it down. Its velocity becomes smaller and smaller until it reaches zero for an instant at the highest point.
At the top of the motion,
$$
v = 0
$$
but the acceleration is still downward:
$$
a = -g
$$
This is a very important idea. Zero velocity does not mean zero acceleration. The object stops only momentarily before starting to move downward.
If an object is launched upward from height $y_0$ with speed $v_0$, then the time to reach the highest point is found from
$$
0 = v_0 - gt
$$
so
$$
t_{\text{top}} = \frac{v_0}{g}
$$
Downward Motion
If an object is dropped, its initial velocity is zero:
$$
v_0 = 0
$$
Then, with upward positive,
$$
v = -gt
$$
and
$$
y = y_0 - \frac{1}{2}gt^2
$$
The negative velocity shows that the object moves downward.
If an object is thrown downward, then its initial velocity is already negative in the upward positive system. Gravity increases the downward speed further.
Interpreting the Signs
The signs in vertical motion carry physical meaning. A positive velocity means motion upward, while a negative velocity means motion downward, if upward is positive. A positive displacement means the final position is above the initial position, while a negative displacement means it is below.
The acceleration due to gravity keeps the same sign during the whole motion because gravity does not reverse direction.
The following table summarizes the meaning of signs when upward is positive.
| Quantity | Positive sign means | Negative sign means |
|---|---|---|
| $y - y_0$ | final position above initial position | final position below initial position |
| $v$ | moving upward | moving downward |
| $a$ | acceleration upward | acceleration downward |
| $g$ in equations | not used alone as sign, usually treated as positive magnitude | sign comes from coordinate choice |
Highest Point in Vertical Motion
The highest point is special because the vertical velocity becomes zero there. This allows us to find the maximum height using the equation
$$
v^2 = v_0^2 - 2g(y - y_0)
$$
At the top, $v = 0$, so
$$
0 = v_0^2 - 2g(y_{\text{max}} - y_0)
$$
which gives
$$
y_{\text{max}} - y_0 = \frac{v_0^2}{2g}
$$
This is the vertical rise above the launch point.
At the highest point of vertical motion,
$$
v = 0
$$
but
$$
a = -g
$$
The maximum rise above the starting point is
$$
\Delta y_{\text{max}} = \frac{v_0^2}{2g}
$$
Symmetry of Upward and Downward Motion
When air resistance is ignored and the object returns to the same height from which it was launched, the motion is symmetric. The time going up equals the time coming down, and the speed when returning equals the launch speed, but the direction is opposite.
So, if an object is thrown upward with speed $v_0$, then when it comes back to the same level,
$$
v = -v_0
$$
if upward is positive.
This symmetry is very useful, but it only applies when the starting and ending heights are the same and air resistance is negligible.
A Simple Example
Imagine a ball thrown straight upward from the ground with initial velocity
$$
v_0 = 19.6\ \text{m/s}
$$
Take upward as positive and use
$$
g = 9.8\ \text{m/s}^2
$$
The time to reach the top is
$$
t_{\text{top}} = \frac{v_0}{g} = \frac{19.6}{9.8} = 2.0\ \text{s}
$$
The maximum height above the ground is
$$
\Delta y_{\text{max}} = \frac{v_0^2}{2g}
= \frac{(19.6)^2}{2(9.8)}
= 19.6\ \text{m}
$$
If the ball returns to the ground, the total time is
$$
t_{\text{total}} = 4.0\ \text{s}
$$
because the upward and downward times are equal.
Visualizing Vertical Motion
A simple sketch helps show the direction of velocity and acceleration during the motion.
Common Mistakes
A frequent mistake is to change the sign of acceleration when the object starts moving downward. The acceleration does not change just because the velocity changes direction. Gravity continues to point downward the whole time.
Another common mistake is to assume that velocity and acceleration must always have the same sign. During upward motion, the velocity is upward but the acceleration is downward, so their signs are opposite.
A third mistake is mixing coordinate choices. If you choose upward as positive, then every position, velocity, displacement, and acceleration must follow that same convention.
Do not change the sign of gravity halfway through a problem. The direction of acceleration is fixed by gravity, and the sign comes only from the coordinate system you chose at the beginning.
Relation to Free Fall
Vertical motion includes free fall as a special case. Free fall usually refers to motion under gravity alone. Vertical motion is broader, because it also includes cases where the object is thrown upward or downward. In all these cases, the same constant acceleration ideas apply.
Final Idea
Vertical motion is one of the clearest examples of constant acceleration. Once the coordinate direction is chosen, the whole problem becomes an exercise in careful sign use. The object may rise, stop momentarily, and fall, but gravity keeps acting downward the entire time. Understanding this makes many later mechanics problems much easier.