Finding displacement when an object is accelerating uniformly can be done using a fundamental equation of motion that relates displacement, initial velocity, acceleration, and time.
Understanding Displacement with Acceleration
Displacement refers to the change in an object's position from its starting point to its ending point, including direction. When an object is accelerating, its velocity is changing over time. To find the displacement in this scenario, you need to consider the initial velocity, the rate of acceleration, and the duration of the motion.
The Key Formula
The primary method for calculating displacement (s) under constant acceleration relies on the following formula:
s = ut + ½at²
This formula is derived from the principles of kinematics, specifically designed for motion with uniform acceleration.
Based on the provided reference, we can define the terms precisely:
Hence, displacement (s) of an object is equal to initial velocity(u) times time (t), plus half of the acceleration (½ a) multiplied by time squared (t²).
Let's break down the components of this formula:
- s: Displacement (often measured in meters, m)
- u: Initial velocity (the object's velocity at the beginning of the time interval, often measured in meters per second, m/s)
- t: Time (the duration over which the motion occurs, often measured in seconds, s)
- a: Acceleration (the rate at which the velocity changes, assumed to be constant, often measured in meters per second squared, m/s²)
Applying the Formula: A Step-by-Step Approach
To use the formula s = ut + ½at², follow these steps:
- Identify the known values: Determine the initial velocity (
u), the acceleration (a), and the time duration (t) from the problem or scenario. - Ensure consistent units: Make sure all your units are compatible (e.g., meters for displacement, meters per second for velocity, meters per second squared for acceleration, and seconds for time).
- Plug the values into the formula: Substitute the known values for
u,a, andtinto the equations = ut + ½at². - Calculate the terms:
- Calculate the product of initial velocity and time (
ut). - Calculate the product of half the acceleration and the square of time (
½at²).
- Calculate the product of initial velocity and time (
- Sum the terms: Add the results from step 4 to find the total displacement (
s).
Example Calculation
Let's consider a simple example:
An object starts moving with an initial velocity of u = 5 m/s. It accelerates uniformly at a rate of a = 2 m/s² for a duration of t = 4 seconds. What is its displacement?
- Knowns:
u = 5 m/s,a = 2 m/s²,t = 4 s. - Units: Units are consistent (meters and seconds).
- Plug in:
s = (5 m/s)(4 s) + ½(2 m/s²)(4 s)² - Calculate terms:
ut = (5 m/s)(4 s) = 20 m½at² = ½(2 m/s²)(16 s²) = (1 m/s²)(16 s²) = 16 m
- Sum terms:
s = 20 m + 16 m = 36 m
Therefore, the displacement of the object after 4 seconds is 36 meters.
Key Variables for Displacement Calculation
| Variable | Symbol | Common Unit | Description |
|---|---|---|---|
| Displacement | s | meters (m) | Change in position (final minus initial) |
| Initial Velocity | u | m/s | Velocity at the start of the time interval |
| Time | t | seconds (s) | Duration of motion |
| Acceleration | a | m/s² | Rate of change of velocity (assumed constant) |
This formula is a powerful tool in physics and engineering for analyzing motion under constant acceleration.
