Determining the initial speed (\( v_1 \)) is crucial in various real-life scenarios, from driving to sports. Knowing how to calculate this using algebraic methods can provide insights into motion and acceleration. Let's explore how to find the initial speed with practical examples.
Formula to Find Initial Speed
The initial speed (\( v_1 \)) can be calculated using 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 \( v_1 \):
\[ v_1 = v_2 - a \cdot t \]
Example 1: Car Braking to a Stop
Scenario: A car traveling at a speed (\( v_2 \)) of \( 15 \, \text{m/s} \) decelerates to a stop (\( v_1 \)) in \( 5 \, \text{seconds} \). The acceleration (\( a \)) is \( -3 \, \text{m/s}^2 \). What was the initial speed?
Step-by-Step Calculation:
1. Given:
\[ v_2 = 15 \, \text{m/s} \]
\[ a = -3 \, \text{m/s}^2 \]
\[ t = 5 \, \text{s} \]
2. Substitute Values into the Initial Speed Formula:
\[ v_1 = v_2 - a \cdot t \]
\[ v_1 = 15 - (-3) \cdot 5 \]
3. Perform the Calculation:
\[ v_1 = 15 + 15 \]
\[ v_1 = 30 \, \text{m/s} \]
Final Value
The initial speed of the car was:
\[ v_1 = 30 \, \text{m/s} \]
Example 2: Sprinter's Start
Scenario: A sprinter reaches a speed (\( v_2 \)) of \( 12 \, \text{m/s} \) after accelerating at \( 4 \, \text{m/s}^2 \) for \( 3 \, \text{seconds} \). What was the initial speed?
Step-by-Step Calculation:
1. Given:
\[ v_2 = 12 \, \text{m/s} \]
\[ a = 4 \, \text{m/s}^2 \]
\[ t = 3 \, \text{s} \]
2. Substitute Values into the Initial Speed Formula:
\[ v_1 = v_2 - a \cdot t \]
\[ v_1 = 12 - 4 \cdot 3 \]
3. Perform the Calculation:
\[ v_1 = 12 - 12 \]
\[ v_1 = 0 \, \text{m/s} \]
Final Value
The initial speed of the sprinter was:
\[ v_1 = 0 \, \text{m/s} \] (starting from rest)
Example 3: Bicycle Deceleration
Scenario: A cyclist is moving at a final speed (\( v_2 \)) of \( 5 \, \text{m/s} \) after decelerating at \( -2 \, \text{m/s}^2 \) for \( 2 \, \text{seconds} \). What was the initial speed?
Step-by-Step Calculation:
1. Given:
\[ v_2 = 5 \, \text{m/s} \]
\[ a = -2 \, \text{m/s}^2 \]
\[ t = 2 \, \text{s} \]
2. Substitute Values into the Initial Speed Formula:
\[ v_1 = v_2 - a \cdot t \]
\[ v_1 = 5 - (-2) \cdot 2 \]
3. Perform the Calculation:
\[ v_1 = 5 + 4 \]
\[ v_1 = 9 \, \text{m/s} \]
Final Value
The initial speed of the cyclist was:
\[ v_1 = 9 \, \text{m/s} \]
Summary
To find the initial speed (\( v_1 \)) when acceleration (\( a \)), final speed (\( v_2 \)), and time (\( t \)) are known, use the formula:
\[ v_1 = v_2 - a \cdot t \]
In the examples provided:
1. A car braking to a stop had an initial speed of \( 30 \, \text{m/s} \).
2. A sprinter accelerating from rest had an initial speed of \( 0 \, \text{m/s} \).
3. A cyclist decelerating had an initial speed of \( 9 \, \text{m/s} \).
This calculation is essential in understanding and analyzing motion in various contexts, from everyday driving to athletic performance.
