Math Problem Statement

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?

Solution

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:

vf=vi+atv_f = v_i + a t

Where:

  • vfv_f is the final velocity (what we are solving for).
  • viv_i is the initial velocity (6.4 m/s).
  • aa is the acceleration (to be determined).
  • tt is the time (converted to seconds).

Step 1: Convert distance and time into consistent units.

  • Distance traveled, d=3.2km=3200md = 3.2 \, \text{km} = 3200 \, \text{m}
  • Time, t=3.5min=3.5×60s=210st = 3.5 \, \text{min} = 3.5 \times 60 \, \text{s} = 210 \, \text{s}

Step 2: Find acceleration using the second kinematic equation:

d=vit+12at2d = v_i t + \frac{1}{2} a t^2

Substitute the known values:

3200=6.4×210+12a×(210)23200 = 6.4 \times 210 + \frac{1}{2} a \times (210)^2

Now, we can solve for aa and then use it to find the final velocity vfv_f. Let's compute this.The final speed of the car after accelerating for 3.5 minutes is approximately 24.08m/s24.08 \, \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!

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Math Problem Analysis

Mathematical Concepts

Kinematics
Uniform Acceleration
Conversion of Units

Formulas

v_f = v_i + a t
d = v_i t + (1/2) a t^2

Theorems

Uniform acceleration equations in kinematics

Suitable Grade Level

Grades 9-12