A car enters the freeway with a speed of 6.4 m/s and accelerates uniformly for 3.2 km in 3.5 min. How fast is the car moving after this time?

To solve this problem, we need to calculate the final speed of the car after it has accelerated uniformly. We'll use the following kinematic equation for uniform acceleration:

\[
v_f = v_i + a t
\]

Where:
- \(v_f\) is the final velocity (what we are solving for).
- \(v_i\) is the initial velocity (6.4 m/s).
- \(a\) is the acceleration (to be determined).
- \(t\) is the time (converted to seconds).

### Step 1: Convert distance and time into consistent units.
- Distance traveled, \(d = 3.2 \, \text{km} = 3200 \, \text{m}\)
- Time, \(t = 3.5 \, \text{min} = 3.5 \times 60 \, \text{s} = 210 \, \text{s}\)

### Step 2: Find acceleration using the second kinematic equation:

\[
d = v_i t + \frac{1}{2} a t^2
\]

Substitute the known values:

\[
3200 = 6.4 \times 210 + \frac{1}{2} a \times (210)^2
\]

Now, we can solve for \(a\) and then use it to find the final velocity \(v_f\). Let's compute this.The final speed of the car after accelerating for 3.5 minutes is approximately \( 24.08 \, \text{m/s} \).

Would you like more details on the solution, or do you have any questions?

Here are some related questions to expand your understanding:
1. How does uniform acceleration affect the distance traveled over time?
2. What would happen if the initial speed was different, say 10 m/s, for the same time and distance?
3. How can you calculate the average speed during the acceleration?
4. What would the acceleration be if the time was doubled but the distance stayed the same?
5. How can you determine the total distance traveled with non-uniform acceleration?

**Tip:** Always ensure units are consistent (e.g., converting time to seconds) when solving physics problems!

Solve the problem of a car entering the freeway with a speed of 6.4 m/s and accelerating uniformly for 3.2 km over 3.5 minutes. Learn how to calculate the final speed using kinematic equations, including uniform acceleration and unit conversion. Suitable for students in Grades 9-12.

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