Linear acceleration is the rate at which an object's velocity changes over a specific period. This article explains how to determine the final speed \( v_2 \) when the initial speed \( v_1 \), acceleration \( a \), and time \( t \) are known.
Formula to Find Final Speed \( v_2 \)
To calculate the final speed \( v_2 \), use the linear acceleration formula rearranged to solve for \( v_2 \):
\[ a = \dfrac{v_2 - v_1}{t} \]
Rearranging to find \( v_2 \):
\[ v_2 = v_1 + a \cdot t \]
Where:
- \( v_1 \) is the initial speed.
- \( v_2 \) is the final speed.
- \( a \) is the linear acceleration.
- \( t \) is the time over which the change in velocity occurs.
Step-by-Step Calculation
Let’s illustrate the calculation of final speed \( v_2 \) with an example.
Example 1: Find Final Speed
Given:
- Initial speed \( v_1 = 10 \, \text{m/s} \)
- Acceleration \( a = 3 \, \text{m/s}^2 \)
- Time \( t = 5 \, \text{s} \)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- Initial speed \( v_1 = 10 \, \text{m/s} \)
- Acceleration \( a = 3 \, \text{m/s}^2 \)
- Time \( t = 5 \, \text{s} \)
Step 2: Substitute the Values into the Final Speed Formula
Using the formula:
\[ v_2 = v_1 + a \cdot t \]
Substitute \( v_1 = 10 \, \text{m/s} \), \( a = 3 \, \text{m/s}^2 \), and \( t = 5 \, \text{s} \):
\[ v_2 = 10 + 3 \cdot 5 \]
Step 3: Calculate the Product of Acceleration and Time
Calculate \( 3 \cdot 5 \):
\[ a \cdot t = 15 \, \text{m/s} \]
Step 4: Add the Product to the Initial Speed
Add to \( v_1 \):
\[ v_2 = 10 + 15 = 25 \, \text{m/s} \]
Final Value
The final speed is \( 25 \, \text{m/s} \).
Example 2: Detailed Calculation
Given:
- Initial speed \( v_1 = 20 \, \text{m/s} \)
- Acceleration \( a = 4 \, \text{m/s}^2 \)
- Time \( t = 3 \, \text{s} \)
Step-by-Step Calculation:
1. Substitute the Given Values into the Formula:
\[ v_2 = v_1 + a \cdot t \]
Given \( v_1 = 20 \, \text{m/s} \), \( a = 4 \, \text{m/s}^2 \), and \( t = 3 \, \text{s} \):
\[ v_2 = 20 + 4 \cdot 3 \]
2. Calculate the Product of Acceleration and Time:
\[ 4 \cdot 3 = 12 \, \text{m/s} \]
3. Add the Product to the Initial Speed:
\[ v_2 = 20 + 12 = 32 \, \text{m/s} \]
Thus, the final speed is \( 32 \, \text{m/s} \).
Let’s consider another example to further illustrate:
Example 3:
Given:
- Initial speed \( v_1 = 5 \, \text{m/s} \)
- Acceleration \( a = 6 \, \text{m/s}^2 \)
- Time \( t = 2 \, \text{s} \)
Calculation:
1. Substitute into the formula:
\[ v_2 = v_1 + a \cdot t \]
Given \( v_1 = 5 \, \text{m/s} \), \( a = 6 \, \text{m/s}^2 \), and \( t = 2 \, \text{s} \):
\[ v_2 = 5 + 6 \cdot 2 \]
2. Calculate the product of acceleration and time:
\[ 6 \cdot 2 = 12 \, \text{m/s} \]
3. Add the product to the initial speed:
\[ v_2 = 5 + 12 = 17 \, \text{m/s} \]
Thus, the final speed is \( 17 \, \text{m/s} \).
Conclusion
Determining the final speed \( v_2 \) using the formula \( v_2 = v_1 + a \cdot t \) is essential for understanding how an object's velocity changes over a given period. This straightforward method helps in predicting the speed of an object at the end of a time interval based on its initial speed, acceleration, and time.
