Question

Distance Traveled over Time by a Car that is Accelerating

Answer 1

To calculate the distance traveled over time by a car that is accelerating, you can use the equations of motion from physics, specifically the equations for uniformly accelerated motion. The key variables involved are:

- \( s \): the distance traveled
- \( u \): the initial velocity (in meters per second, m/s)
- \( a \): the acceleration (in meters per second squared, m/s²)
- \( t \): the time the car has been accelerating (in seconds, s)

The primary equation for the distance traveled under constant acceleration is:

\[
s = ut + \frac{1}{2} a t^2
\]

### Variables Explained:

1. **Initial Velocity (\( u \))**: This is the speed of the car at the beginning of the time interval (if the car is starting from rest, \( u = 0 \)).
2. **Acceleration (\( a \))**: This is the rate at which the car's speed increases.
3. **Time (\( t \))**: The total time over which the car has been accelerating.

### Example Calculation:

Let's say:

- The initial velocity (\( u \)) of the car is 0 m/s (the car starts from rest).
- The acceleration (\( a \)) is 2 m/s².
- The time (\( t \)) is 5 seconds.

Substituting these values into the equation:

\[
s = 0 \cdot 5 + \frac{1}{2} \cdot 2 \cdot (5^2)
\]
\[
s = 0 + \frac{1}{2} \cdot 2 \cdot 25
\]
\[
s = 0 + 25 = 25 \text{ meters}
\]

So, the car travels 25 meters in 5 seconds while accelerating from rest at a rate of 2 m/s².

### Additional Considerations:

- If the initial velocity is not zero, simply plug in that value for \( u \).
- If the acceleration is not constant, the situation becomes more complex and may require calculus to solve.
- Make sure that all units are consistent when performing calculations (e.g., meters for distance and seconds for time).

If you have a specific situation or values in mind, please provide those for a more tailored example!