Calculating the final speed (\( v_2 \)) is crucial in many real-world situations, such as determining how fast a car will be moving after a period of acceleration or understanding the final speed of an object in motion. This article provides a step-by-step guide to finding the final speed using algebraic methods, with practical examples.
Formula to Find Final Speed
The final speed (\( v_2 \)) can be derived from the formula for acceleration (\( a \)):
\[ a = \dfrac{v_2 - v_1}{t} \]
where:
- \( a \) is the acceleration.
- \( v_2 \) is the final speed.
- \( v_1 \) is the initial speed.
- \( t \) is the time taken.
Rearrange this formula to solve for \( v_2 \):
\[ v_2 = v_1 + a \cdot t \]
Example 1: Speeding Up on the Highway
Scenario: A car starts at a speed (\( v_1 \)) of \( 20 \, \text{m/s} \) and accelerates at \( 4 \, \text{m/s}^2 \) for \( 5 \, \text{seconds} \). What is the final speed?
Step-by-Step Calculation:
1. Given:
\[ v_1 = 20 \, \text{m/s} \]
\[ a = 4 \, \text{m/s}^2 \]
\[ t = 5 \, \text{s} \]
2. Substitute Values into the Final Speed Formula:
\[ v_2 = v_1 + a \cdot t \]
\[ v_2 = 20 + 4 \cdot 5 \]
3. Perform the Calculation:
\[ v_2 = 20 + 20 \]
\[ v_2 = 40 \, \text{m/s} \]
Final Value
The final speed of the car is:
\[ v_2 = 40 \, \text{m/s} \]
Example 2: Cyclist Accelerating
Scenario: A cyclist starts from rest (\( v_1 \)) and accelerates at \( 3 \, \text{m/s}^2 \) for \( 6 \, \text{seconds} \). What is the final speed?
Step-by-Step Calculation:
1. Given:
\[ v_1 = 0 \, \text{m/s} \]
\[ a = 3 \, \text{m/s}^2 \]
\[ t = 6 \, \text{s} \]
2. Substitute Values into the Final Speed Formula:
\[ v_2 = v_1 + a \cdot t \]
\[ v_2 = 0 + 3 \cdot 6 \]
3. Perform the Calculation:
\[ v_2 = 0 + 18 \]
\[ v_2 = 18 \, \text{m/s} \]
Final Value
The final speed of the cyclist is:
\[ v_2 = 18 \, \text{m/s} \]
Example 3: Object Slowing Down
Scenario: An object is moving with an initial speed (\( v_1 \)) of \( 15 \, \text{m/s} \) and decelerates at \( -2 \, \text{m/s}^2 \) for \( 4 \, \text{seconds} \). What is the final speed?
Step-by-Step Calculation:
1. Given:
\[ v_1 = 15 \, \text{m/s} \]
\[ a = -2 \, \text{m/s}^2 \]
\[ t = 4 \, \text{s} \]
2. Substitute Values into the Final Speed Formula:
\[ v_2 = v_1 + a \cdot t \]
\[ v_2 = 15 + (-2) \cdot 4 \]
3. Perform the Calculation:
\[ v_2 = 15 - 8 \]
\[ v_2 = 7 \, \text{m/s} \]
Final Value
The final speed of the object is:
\[ v_2 = 7 \, \text{m/s} \]
Summary
To find the final speed (\( v_2 \)) when acceleration (\( a \)), initial speed (\( v_1 \)), and time (\( t \)) are known, use the formula:
\[ v_2 = v_1 + a \cdot t \]
In the examples provided:
1. A car speeding up on the highway reaches a final speed of \( 40 \, \text{m/s} \).
2. A cyclist accelerating from rest reaches a final speed of \( 18 \, \text{m/s} \).
3. An object slowing down has a final speed of \( 7 \, \text{m/s} \).
This calculation is essential for analyzing and predicting the motion of objects in various practical applications.
