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To analyze the motion of a car traveling in a straight line with a varying velocity defined by the equation \( v = |t - t_0| \) m/s for the time interval \( 0 \leq t \leq 2t_0 \), we need to break down the components of this equation and understand how the car's velocity changes over time.

Understanding the Velocity Function

The velocity function \( v = |t - t_0| \) indicates that the velocity depends on the absolute difference between the current time \( t \) and a specific time \( t_0 \). This means that:

• For \( t < t_0 \), the expression \( t - t_0 \) is negative, so \( v = -(t - t_0) = t_0 - t \).
• For \( t = t_0 \), the velocity \( v = 0 \) m/s, indicating that this is the moment when the car stops.
• For \( t > t_0 \), the expression \( t - t_0 \) is positive, so \( v = t - t_0 \).

Velocity Over Time

Now, let's visualize how the velocity changes over the specified time interval:

• From \( t = 0 \) to \( t = t_0 \): The velocity increases linearly from \( t_0 \) to \( 0 \). For example, if \( t_0 = 5 \) seconds, then at \( t = 0 \), \( v = 5 \) m/s, and at \( t = 5 \), \( v = 0 \) m/s.
• At \( t = t_0 \): The car is at rest, with a velocity of \( 0 \) m/s.
• From \( t = t_0 \) to \( t = 2t_0 \): The velocity increases again, starting from \( 0 \) m/s at \( t_0 \) and reaching \( t_0 \) m/s at \( t = 2t_0 \). Continuing the previous example, at \( t = 10 \) seconds, \( v = 5 \) m/s.

Distance Traveled

To find the distance traveled by the car during this time, we can integrate the velocity function over the specified intervals. The total distance \( s \) can be calculated as follows:

Calculating Distance in Each Interval

1. **From \( 0 \) to \( t_0 \)**:

Here, the velocity function is \( v = t_0 - t \). The distance \( s_1 \) can be calculated using the integral:

s_1 = ∫(t_0 - t) dt from 0 to t_0

Evaluating this integral gives:

s_1 = [t_0 t - (t^2)/2] from 0 to t_0 = t_0^2/2.

2. **From \( t_0 \) to \( 2t_0 \)**:

In this interval, the velocity function is \( v = t - t_0 \). The distance \( s_2 \) can be calculated as:

s_2 = ∫(t - t_0) dt from t_0 to 2t_0

Evaluating this integral gives:

s_2 = [(t^2)/2 - t_0 t] from t_0 to 2t_0 = (2t_0^2)/2 - t_0(2t_0) - (t_0^2/2 - t_0^2) = t_0^2/2.

Total Distance

Finally, the total distance \( s \) traveled by the car during the entire time interval is:

s = s_1 + s_2 = (t_0^2/2) + (t_0^2/2) = t_0^2.

This analysis shows that the car travels a total distance of \( t_0^2 \) meters during the time interval from \( 0 \) to \( 2t_0 \). Understanding how to interpret the velocity function and calculate the distance using integration is crucial in kinematics, as it allows us to analyze motion in a more comprehensive way.