How to Calculate Vertical Velocity With Formulas and Examples

Understanding how objects move vertically, whether rising or falling, is a fundamental aspect of physics. Analyzing vertical motion allows for the prediction of an object’s path and speed. This article explains how to calculate vertical velocity.

What is Vertical Velocity?

Vertical velocity refers to the speed of an object moving directly upwards or downwards, perpendicular to the Earth’s surface. It quantifies the rate at which an object’s vertical position changes. As a vector quantity, it possesses both a numerical value and a specific direction. Upward movement is typically positive, while downward movement is negative. This directional component distinguishes it from overall speed or horizontal velocity.

Essential Concepts for Calculation

Calculating vertical velocity relies on understanding several fundamental concepts. Initial velocity (v₀) represents the speed and direction an object has at the beginning of its vertical motion. This starting velocity is a critical input for predicting future motion. Acceleration due to gravity (‘g’) is a constant force pulling objects towards the Earth’s center, approximately 9.8 meters per second squared (m/s²). This means an object’s downward velocity increases by 9.8 m/s each second it is in free fall. Time (‘t’) is the duration an object is in motion. Displacement (‘d’ or ‘s’) refers to the change in an object’s vertical position from its starting point. For example, if an object moves from 10 meters high to 5 meters high, its displacement is -5 meters. These variables form the basis for vertical motion equations.

Core Formulas for Vertical Velocity

Several fundamental kinematic equations calculate vertical velocity.
One primary formula is v = v₀ + at, where ‘v’ is the final vertical velocity, ‘v₀’ is the initial vertical velocity, ‘a’ is the acceleration, and ‘t’ is the time. In vertical motion, ‘a’ is replaced by ‘g’, the acceleration due to gravity.
Another equation is d = v₀t + ½at², which determines displacement based on initial velocity, acceleration, and time.
When finding the final velocity without knowing the time, the formula v² = v₀² + 2ad applies. These equations allow for calculating an object’s vertical velocity at any moment.

Step-by-Step Calculation Examples

To illustrate the application of these formulas, consider a few common scenarios.

Example 1: Object Dropped from Rest

Imagine dropping a ball from a tall building. The problem asks for the ball’s vertical velocity after 3 seconds.
Knowns: Initial vertical velocity (v₀) = 0 m/s (since it’s dropped from rest), acceleration due to gravity (g) = 9.8 m/s², time (t) = 3 s.
Formula: v = v₀ + gt
Calculation: v = 0 m/s + (9.8 m/s²)(3 s) = 29.4 m/s.
The final vertical velocity of the ball after 3 seconds is 29.4 m/s downwards.

Example 2: Object Thrown Upwards

Suppose a ball is thrown straight upwards with an initial velocity of 15 m/s. We want to find its vertical velocity after 1 second.
Knowns: Initial vertical velocity (v₀) = +15 m/s (positive for upward motion), acceleration due to gravity (g) = -9.8 m/s² (negative because gravity acts downwards), time (t) = 1 s.
Formula: v = v₀ + gt
Calculation: v = 15 m/s + (-9.8 m/s²)(1 s) = 15 m/s – 9.8 m/s = 5.2 m/s.
After 1 second, the ball’s vertical velocity is 5.2 m/s upwards, indicating it is still rising but slowing down.

Example 3: Object Falling for a Given Displacement

Consider an object that falls 20 meters from rest. What is its final vertical velocity just before it hits the ground?
Knowns: Initial vertical velocity (v₀) = 0 m/s, displacement (d) = -20 m (negative for downward displacement), acceleration due to gravity (g) = -9.8 m/s².
Formula: v² = v₀² + 2gd
Calculation: v² = (0 m/s)² + 2(-9.8 m/s²)(-20 m) = 0 + 392 = 392.
v = √392 ≈ 19.8 m/s.
The final vertical velocity of the object is approximately 19.8 m/s downwards.

Factors Influencing Vertical Motion

Several factors significantly influence an object’s vertical motion and its calculated velocity.
The most prominent is the constant acceleration due to gravity. This force uniformly pulls objects downwards, causing their vertical velocity to change consistently, whether they are moving upwards or falling. This downward acceleration means an object thrown upwards will slow down, momentarily stop at its peak, and then accelerate downwards.
The initial velocity of an object also plays a substantial role. The magnitude and direction of this starting velocity determine how high an object will travel, its total flight time, and its velocity at various points in its trajectory. For instance, a higher initial upward velocity will result in a greater maximum height.
While often disregarded in basic calculations for simplicity, air resistance is another real-world factor. It acts as a drag force opposing motion, reducing an object’s speed, especially at higher velocities.