Linear acceleration measures how quickly an object's velocity changes over time. Sometimes, you may need to find the initial speed \( v_1 \) when the final speed \( v_2 \), acceleration \( a \), and time \( t \) are known. This article will guide you through calculating the initial speed \( v_1 \) using a simple algebraic formula.
Formula to Find Initial Speed \( v_1 \)
The formula to calculate the initial speed \( v_1 \) is derived from the linear acceleration equation:
\[ a = \dfrac{v_2 - v_1}{t} \]
Rearrange the formula to solve for \( v_1 \):
\[ v_1 = v_2 - a \cdot t \]
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 initial speed \( v_1 \) with an example:
Given:
- Final speed \( v_2 = 50 \, \text{m/s} \)
- Acceleration \( a = 4 \, \text{m/s}^2 \)
- Time \( t = 5 \, \text{s} \)
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- Final speed \( v_2 = 50 \, \text{m/s} \)
- Acceleration \( a = 4 \, \text{m/s}^2 \)
- Time \( t = 5 \, \text{s} \)
Step 2: Substitute the Values into the Initial Speed Formula
Using the formula:
\[ v_1 = v_2 - a \cdot t \]
Substitute \( v_2 = 50 \, \text{m/s} \), \( a = 4 \, \text{m/s}^2 \), and \( t = 5 \, \text{s} \):
\[ v_1 = 50 - 4 \cdot 5 \]
Step 3: Calculate the Product of Acceleration and Time
Calculate \( 4 \cdot 5 \):
\[ a \cdot t = 20 \, \text{m/s} \]
Step 4: Subtract the Product from the Final Speed
Subtract from \( v_2 \):
\[ v_1 = 50 - 20 = 30 \, \text{m/s} \]
Final Value
The initial speed is \( 30 \, \text{m/s} \).
Additional Example
Let's consider another example for clarity:
Example 2:
- Final speed \( v_2 = 80 \, \text{m/s} \)
- Acceleration \( a = 5 \, \text{m/s}^2 \)
- Time \( t = 6 \, \text{s} \)
Calculation:
1. Substitute into the formula:
\[ v_1 = v_2 - a \cdot t \]
Given \( v_2 = 80 \, \text{m/s} \), \( a = 5 \, \text{m/s}^2 \), and \( t = 6 \, \text{s} \):
\[ v_1 = 80 - 5 \cdot 6 \]
2. Calculate the product of acceleration and time:
\[ 5 \cdot 6 = 30 \, \text{m/s} \]
3. Subtract the product from the final speed:
\[ v_1 = 80 - 30 = 50 \, \text{m/s} \]
Thus, the initial speed is \( 50 \, \text{m/s} \).
Conclusion
Calculating the initial speed \( v_1 \) using the formula \( v_1 = v_2 - a \cdot t \) is essential for understanding the starting conditions of motion in various physical scenarios. This straightforward method helps determine the speed of an object at the beginning of a given time interval, based on its final speed, acceleration, and time.
