When the initial velocity is zero, calculating the distance traveled given the acceleration and time becomes straightforward. This scenario is common in everyday situations such as a car starting from rest or a cyclist beginning to pedal.
Distance Formula
The formula to calculate the distance (\( d \)) when the initial velocity (\( v_1 \)) is zero is:
\[ d = \dfrac{1}{2} \cdot a \cdot t^2 \]
Where:
- \( t \) is the time
- \( a \) is the acceleration
Explanation of the Formula
\( \dfrac{1}{2} \cdot a \cdot t^2 \): This term represents the distance traveled due to constant acceleration over the time \( t \). Since the initial velocity is zero, there is no initial distance contribution, simplifying the equation to just the acceleration term.
Step-by-Step Calculation
Let's work through an example to illustrate how to use this formula in a real-life scenario.
Example: Calculating the Distance for a Car Starting from Rest
1. Scenario: A car starts from rest at a traffic light. The car accelerates uniformly at a rate of 3 m/s² for a duration of 10 seconds. We need to find the distance the car travels in this time.
2. Identify the given values:
- Time (\( t \)) = 10 seconds
- Acceleration (\( a \)) = 3 m/s²
3. Substitute the values into the distance formula:
\[ d = \dfrac{1}{2} \cdot 3 \cdot 10^2 \]
4. Calculate the term (\( \dfrac{1}{2} \cdot a \cdot t^2 \)):
\[ \dfrac{1}{2} \cdot 3 \cdot 10^2 = \dfrac{1}{2} \cdot 3 \cdot 100 = \dfrac{3 \cdot 100}{2} = 150 \]
So, the total distance traveled by the car is 150 meters.
This example shows how you can calculate the distance traveled by a car starting from rest and accelerating uniformly. The same formula can be applied to other real-life scenarios, such as a cyclist beginning to pedal or an object falling from rest under gravity. By understanding and applying this formula, you can solve various practical motion problems where the initial velocity is zero.
