Determining the linear acceleration \( a \) when the distance \( d \), initial speed \( v_1 \), and time \( t \) are known is crucial in various fields, including physics and engineering. This article explains how to find \( a \) by rearranging the distance formula and provides examples with step-by-step calculations.
Formula to Find Linear Acceleration \( a \)
To calculate linear acceleration \( a \), we start with the distance formula:
\[ d = v_1 \cdot t + \dfrac{1}{2} \cdot a \cdot t^2 \]
Rearranging the formula to solve for \( a \):
\[ a = \dfrac{2 \cdot (d - v_1 \cdot t)}{t^2} \]
Where:
- \( d \) is the distance traveled.
- \( v_1 \) is the initial speed.
- \( t \) is the time interval.
- \( a \) is the linear acceleration.
Step-by-Step Calculation
Let’s illustrate the calculation of linear acceleration \( a \) with examples.
Example 1: Find Linear Acceleration
Given:
- Distance \( d = 100 \, \text{m} \)
- Initial speed \( v_1 = 10 \, \text{m/s} \)
- Time \( t = 5 \, \text{s} \)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- Distance \( d = 100 \, \text{m} \)
- Initial speed \( v_1 = 10 \, \text{m/s} \)
- Time \( t = 5 \, \text{s} \)
Step 2: Substitute the Values into the Acceleration Formula
Using the formula:
\[ a = \dfrac{2 \cdot (d - v_1 \cdot t)}{t^2} \]
Substitute \( d = 100 \, \text{m} \), \( v_1 = 10 \, \text{m/s} \), and \( t = 5 \, \text{s} \):
\[ a = \dfrac{2 \cdot (100 - 10 \cdot 5)}{5^2} \]
Step 3: Calculate the Distance Traveled by Initial Speed
Calculate \( v_1 \cdot t \):
\[ v_1 \cdot t = 10 \cdot 5 = 50 \, \text{m} \]
Step 4: Subtract This Value from Total Distance
Subtract \( 50 \) from \( 100 \):
\[ d - v_1 \cdot t = 100 - 50 = 50 \, \text{m} \]
Step 5: Multiply by 2
Multiply \( 50 \) by \( 2 \):
\[ 2 \cdot 50 = 100 \, \text{m} \]
Step 6: Divide by \( t^2 \)
Divide by \( 25 \):
\[ a = \dfrac{100}{25} = 4 \, \text{m/s}^2 \]
Final Value
The linear acceleration is \( 4 \, \text{m/s}^2 \).
Example 2
Let's break down another calculation for clarity.
Given:
- Distance \( d = 150 \, \text{m} \)
- Initial speed \( v_1 = 0 \, \text{m/s} \)
- Time \( t = 10 \, \text{s} \)
Step-by-Step Calculation:
1. Substitute the Given Values into the Formula:
\[ a = \dfrac{2 \cdot (d - v_1 \cdot t)}{t^2} \]
Given \( d = 150 \, \text{m} \), \( v_1 = 0 \, \text{m/s} \), and \( t = 10 \, \text{s} \):
\[ a = \dfrac{2 \cdot (150 - 0 \cdot 10)}{10^2} \]
2. Calculate the Distance Traveled by Initial Speed:
Since \( v_1 = 0 \), \( v_1 \cdot t = 0 \).
3. Subtract This Value from Total Distance:
\[ d - v_1 \cdot t = 150 - 0 = 150 \, \text{m} \]
4. Multiply by 2:
\[ 2 \cdot 150 = 300 \, \text{m} \]
5. Divide by \( t^2 \):
\[ a = \dfrac{300}{100} = 3 \, \text{m/s}^2 \]
Thus, the linear acceleration is \( 3 \, \text{m/s}^2 \).
Additional Example
Let’s consider another example to further illustrate:
Example 3:
Given:
- Distance \( d = 80 \, \text{m} \)
- Initial speed \( v_1 = 5 \, \text{m/s} \)
- Time \( t = 8 \, \text{s} \)
Calculation:
1. Substitute into the formula:
\[ a = \dfrac{2 \cdot (d - v_1 \cdot t)}{t^2} \]
Given \( d = 80 \, \text{m} \), \( v_1 = 5 \, \text{m/s} \), and \( t = 8 \, \text{s} \):
\[ a = \dfrac{2 \cdot (80 - 5 \cdot 8)}{8^2} \]
2. Calculate the distance traveled by initial speed:
\[ v_1 \cdot t = 5 \cdot 8 = 40 \, \text{m} \]
3. Subtract this value from total distance:
\[ d - v_1 \cdot t = 80 - 40 = 40 \, \text{m} \]
4. Multiply by 2:
\[ 2 \cdot 40 = 80 \, \text{m} \]
5. Divide by \( t^2 \):
\[ a = \dfrac{80}{64} = 1.25 \, \text{m/s}^2 \]
Thus, the linear acceleration is \( 1.25 \, \text{m/s}^2 \).
Conclusion
Using the formula \( a = \dfrac{2 \cdot (d - v_1 \cdot t)}{t^2} \), you can calculate linear acceleration when distance, initial speed, and time are known. This method helps in understanding how quickly an object's velocity changes over time, which is fundamental in analyzing motion in physics and engineering.
