Calculating the distance traveled when the initial speed, acceleration, and time are known is a fundamental problem in kinematics. Using the given parameters, you can find the distance using the following formula:
Distance Formula
The formula to calculate the distance (\( d \)) is:
\[ d = v_1 \cdot t + \dfrac{1}{2} \cdot a \cdot t^2 \]
Where:
- \( v_1 \) is the initial speed
- \( t \) is the time
- \( a \) is the acceleration
Explanation of the Formula
- \( v_1 \cdot t \): This term represents the distance traveled at a constant speed \( v_1 \) over the time \( t \).
- \( \dfrac{1}{2} \cdot a \cdot t^2 \): This term accounts for the additional distance traveled due to the acceleration over the time \( t \).
Step-by-Step Calculation
Let's work through an example to illustrate how to use this formula.
Example: Calculating the Distance
1. Identify the given values:
- Initial speed (\( v_1 \)) = 10 m/s
- Time (\( t \)) = 5 seconds
- Acceleration (\( a \)) = 2 m/s²
2. Substitute the values into the distance formula:
\[ d = 10 \cdot 5 + \dfrac{1}{2} \cdot 2 \cdot 5^2 \]
3. Calculate the first term (\( v_1 \cdot t \)):
\[ 10 \cdot 5 = 50 \]
4. Calculate the second term (\( \dfrac{1}{2} \cdot a \cdot t^2 \)):
\[ \dfrac{1}{2} \cdot 2 \cdot 5^2 = 1 \cdot 25 = 25 \]
5. Add the two terms to find the total distance:
\[ d = 50 + 25 = 75 \]
So, the total distance traveled is 75 meters.
By following these steps, you can easily calculate the distance traveled when the initial speed, acceleration, and time are known. This formula is crucial for solving problems related to motion in physics and engineering.
