Linear acceleration is a fundamental concept in physics that describes the rate of change of velocity of an object in a straight line. Understanding how to calculate linear acceleration is crucial for analyzing motion in various scientific and engineering applications. This article will guide you through the process of calculating linear acceleration using a simple algebraic formula.
Formula for Linear Acceleration
The formula to find linear acceleration \( a \) is:
\[ a = \dfrac{v_2 - v_1}{t} \]
Where:
- \( a \) is the linear acceleration.
- \( v_2 \) is the final velocity.
- \( v_1 \) is the initial velocity.
- \( t \) is the time over which the change in velocity occurs.
Step-by-Step Calculation
Let's illustrate the calculation of linear acceleration with an example:
Given:
- Initial velocity \( v_1 = 10 \, \text{m/s} \)
- Final velocity \( v_2 = 30 \, \text{m/s} \)
- Time \( t = 5 \, \text{s} \)
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- Initial velocity \( v_1 = 10 \, \text{m/s} \)
- Final velocity \( v_2 = 30 \, \text{m/s} \)
- Time \( t = 5 \, \text{s} \)
Step 2: Substitute the Values into the Linear Acceleration Formula
Using the formula:
\[ a = \dfrac{v_2 - v_1}{t} \]
Substitute \( v_2 = 30 \, \text{m/s} \), \( v_1 = 10 \, \text{m/s} \), and \( t = 5 \, \text{s} \):
\[ a = \dfrac{30 - 10}{5} \]
Step 3: Calculate the Difference in Velocity
Calculate \( 30 - 10 \):
\[ v_2 - v_1 = 20 \, \text{m/s} \]
Step 4: Divide by the Time
Divide by \( t \):
\[ a = \dfrac{20}{5} = 4 \, \text{m/s}^2 \]
Final Value
The linear acceleration is \( 4 \, \text{m/s}^2 \).
Detailed Example Calculation
Let's break down the calculation further:
1. Substitute the Given Values into the Formula:
\[ a = \dfrac{v_2 - v_1}{t} \]
Given \( v_1 = 10 \, \text{m/s} \), \( v_2 = 30 \, \text{m/s} \), and \( t = 5 \, \text{s} \):
\[ a = \dfrac{30 - 10}{5} \]
2. Calculate the Difference in Velocity:
\[ 30 - 10 = 20 \, \text{m/s} \]
3. Divide the Change in Velocity by the Time:
\[ a = \dfrac{20}{5} = 4 \, \text{m/s}^2 \]
The calculated linear acceleration is \( 4 \, \text{m/s}^2 \).
Additional Example
Let's consider another example for clarity:
Example 2:
- Initial velocity \( v_1 = 15 \, \text{m/s} \)
- Final velocity \( v_2 = 45 \, \text{m/s} \)
- Time \( t = 10 \, \text{s} \)
Calculation:
1. Substitute into the formula:
\[ a = \dfrac{v_2 - v_1}{t} \]
Given \( v_1 = 15 \, \text{m/s} \), \( v_2 = 45 \, \text{m/s} \), and \( t = 10 \, \text{s} \):
\[ a = \dfrac{45 - 15}{10} \]
2. Calculate the difference in velocity:
\[ 45 - 15 = 30 \, \text{m/s} \]
3. Divide by the time:
\[ a = \dfrac{30}{10} = 3 \, \text{m/s}^2 \]
Thus, the linear acceleration is \( 3 \, \text{m/s}^2 \).
Conclusion
Calculating linear acceleration using the formula \( a = \dfrac{v_2 - v_1}{t} \) is essential for understanding motion in various physical contexts. This straightforward method helps determine how quickly an object's velocity changes over time, which is a key parameter in dynamics and kinematics.
