Understanding how to determine the time interval \( t \) when the initial speed \( v_1 \), final speed \( v_2 \), and acceleration \( a \) are known is essential in various applications. This article explains how to find \( t \) with step-by-step calculations and examples.
Formula to Find Time Interval \( t \)
To calculate the time interval \( t \), use the linear acceleration formula rearranged to solve for \( t \):
\[ a = \dfrac{v_2 - v_1}{t} \]
Rearranging to find \( t \):
\[ t = \dfrac{v_2 - v_1}{a} \]
Where:
- \( v_1 \) is the initial speed.
- \( v_2 \) is the final speed.
- \( a \) is the linear acceleration.
- \( t \) is the time interval over which the change in velocity occurs.
Step-by-Step Calculation
Let’s illustrate the calculation of time interval \( t \) with examples.
Example 1: Find Time Interval
Given:
- Initial speed \( v_1 = 20 \, \text{m/s} \)
- Final speed \( v_2 = 50 \, \text{m/s} \)
- Acceleration \( a = 6 \, \text{m/s}^2 \)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- Initial speed \( v_1 = 20 \, \text{m/s} \)
- Final speed \( v_2 = 50 \, \text{m/s} \)
- Acceleration \( a = 6 \, \text{m/s}^2 \)
Step 2: Substitute the Values into the Time Formula
Using the formula:
\[ t = \dfrac{v_2 - v_1}{a} \]
Substitute \( v_1 = 20 \, \text{m/s} \), \( v_2 = 50 \, \text{m/s} \), and \( a = 6 \, \text{m/s}^2 \):
\[ t = \dfrac{50 - 20}{6} \]
Step 3: Calculate the Difference in Speeds
Calculate \( 50 - 20 \):
\[ v_2 - v_1 = 30 \, \text{m/s} \]
Step 4: Divide by Acceleration
Divide by \( a \):
\[ t = \dfrac{30}{6} = 5 \, \text{s} \]
Final Value
The time interval is \( 5 \, \text{s} \).
Let's break down another calculation for clarity.
Example 2: Detailed Calculation
Given:
- Initial speed \( v_1 = 10 \, \text{m/s} \)
- Final speed \( v_2 = 25 \, \text{m/s} \)
- Acceleration \( a = 3 \, \text{m/s}^2 \)
Step-by-Step Calculation:
1. Substitute the Given Values into the Formula:
\[ t = \dfrac{v_2 - v_1}{a} \]
Given \( v_1 = 10 \, \text{m/s} \), \( v_2 = 25 \, \text{m/s} \), and \( a = 3 \, \text{m/s}^2 \):
\[ t = \dfrac{25 - 10}{3} \]
2. Calculate the Difference in Speeds:
\[ 25 - 10 = 15 \, \text{m/s} \]
3. Divide by Acceleration:
\[ t = \dfrac{15}{3} = 5 \, \text{s} \]
Thus, the time interval is \( 5 \, \text{s} \).
Additional Example
Let’s consider another example to further illustrate:
Example 3:
Given:
- Initial speed \( v_1 = 30 \, \text{m/s} \)
- Final speed \( v_2 = 90 \, \text{m/s} \)
- Acceleration \( a = 10 \, \text{m/s}^2 \)
Calculation:
1. Substitute into the formula:
\[ t = \dfrac{v_2 - v_1}{a} \]
Given \( v_1 = 30 \, \text{m/s} \), \( v_2 = 90 \, \text{m/s} \), and \( a = 10 \, \text{m/s}^2 \):
\[ t = \dfrac{90 - 30}{10} \]
2. Calculate the difference in speeds:
\[ 90 - 30 = 60 \, \text{m/s} \]
3. Divide by acceleration:
\[ t = \dfrac{60}{10} = 6 \, \text{s} \]
Thus, the time interval is \( 6 \, \text{s} \).
Conclusion
Determining the time interval \( t \) using the formula \( t = \dfrac{v_2 - v_1}{a} \) is crucial for understanding the duration over which an object's velocity changes due to acceleration. This method provides an accurate way to calculate how long it takes for an object to reach a certain speed from its initial speed under a given acceleration.
