Vertical Motion

A Vertical Axis Needs a Positive Direction

Vertical motion is upward or downward motion along one straight line. To keep the signs clear, we choose upward as positive and downward as negative.

In the near-Earth free-fall model, gravitational acceleration is written as $-g$ because it points downward with this sign convention.

y=y_0+v_0t-\frac{1}{2}gt^2

These equations model motion near Earth's surface with constant $g$ and no air resistance. The standard value is $g_n=9.80665\text{ m/s}^2$ . The calculations below use $g=10\text{ m/s}^2$ to make the arithmetic easier to do by hand.

Ball in Vertical Motion

Choose how the ball starts moving to see how gravity changes its height and velocity.

Initial condition: $v_0=20\text{ m/s},\quad y_0=0\text{ m}$
Highest point: $20\text{ m}$
Motion time: $4\text{ s}$
Final velocity: $-20\text{ m/s}$

Gravity Still Acts While the Object Rises

When an object is thrown upward, its initial velocity is positive. After that, gravity reduces its vertical velocity little by little until the object stops for an instant at the highest point.

At the highest point, the vertical velocity is momentarily zero, but gravitational acceleration still points downward.

Gravity does not disappear at the top. At that instant, the vertical velocity is zero while the acceleration remains $-g$ .

An Upward Throw Stops for an Instant

Suppose a ball is thrown upward with initial velocity $20\text{ m/s}$ from position $y_0=0$ . Use $g=10\text{ m/s}^2$ . At the highest point, its vertical velocity is $v_y=0$ .

v_y=v_0-gt

The maximum height above the launch point can be found from the position equation.

\begin{aligned} y-y_0&=v_0t-\frac{1}{2}gt^2\\ &=20(2)-\frac{1}{2}(10)(2^2)\\ &=20\text{ m} \end{aligned}

So the ball takes $2\text{ s}$ to reach the top and rises $20\text{ m}$ above its starting point.

Reading the Sign of Velocity

The sign of $v_y$ tells the direction of motion at that moment. A positive value means the object is moving upward, a negative value means it is moving downward, and zero means it is momentarily stopped.

Notice that height $y$ does not automatically tell the motion direction. An object can be at the same height while moving upward and while moving downward, but the velocity signs are different.

Value of $v_y$	Motion Direction	Example Situation
Positive	Upward	A ball just thrown upward
Zero	Stopped for an instant	A ball at the highest point
Negative	Downward	A ball falling back down

A Ball Dropped From a Height

Suppose a ball is dropped from a height of $20\text{ m}$ with no initial velocity. With $g=10\text{ m/s}^2$ , the time to reach the ground is:

\begin{aligned} 0&=20-\frac{1}{2}(10)t^2\\ 5t^2&=20\\ t&=2\text{ s} \end{aligned}

The velocity just before reaching the ground is:

v_y=0-10(2)=-20\text{ m/s}

The negative sign shows the downward direction. So the speed is $20\text{ m/s}$ , but the motion direction is downward.

This example shows why the sign in vertical motion cannot be ignored. The speed tells how fast the ball is moving, while the sign tells the direction of its motion.

From Vertical Motion to Projectile Motion

Vertical motion is also one component of projectile motion. When an object is launched at an angle, its horizontal velocity remains constant, while its vertical velocity changes according to the gravity equations used above.

The same free-fall equations describe the vertical component of a thrown ball, a projectile, or water leaving a hose because gravity supplies the vertical acceleration.