a car travel in straight line with varying velocity given by v=[t-t₀]([ is a modulus)ms^-1 for00..where t is time & t₀ is the time where velocity is zero..

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, we need to break down the components of this equation and understand what it signifies in terms of the car's movement over time.

Understanding the Velocity Function

The expression \( v = |t - t_0| \) indicates that the velocity of the car depends on the time \( t \) and a specific moment \( t_0 \) when the velocity is zero. The modulus function, or absolute value, means that the velocity will always be non-negative, regardless of whether \( t \) is greater than or less than \( t_0 \).

Behavior of the Velocity

Let's analyze how the velocity behaves over time:

• When \( t < t_0 \): The expression \( t - t_0 \) is negative, so \( v = |t - t_0| = -(t - t_0) = t_0 - t \).
• When \( t = t_0 \): The velocity \( v = |t - t_0| = 0 \). This is the point where the car is momentarily at rest.
• When \( t > t_0 \): The expression \( t - t_0 \) is positive, so \( v = |t - t_0| = t - t_0 \).

Graphical Representation

Visualizing this function can help solidify your understanding. If you were to plot \( v \) against \( t \), you would see a V-shaped graph:

• The left arm of the V (for \( t < t_0 \)) slopes upwards as \( t \) approaches \( t_0 \) from the left.
• The point at \( t_0 \) is the vertex of the V, where the velocity is zero.
• The right arm of the V (for \( t > t_0 \)) slopes upwards as \( t \) increases, indicating that the car is accelerating away from the point of rest.

Distance Traveled

To find the distance traveled by the car over a given time interval, we need to integrate the velocity function. The distance \( s \) can be calculated as:

• For \( t < t_0 \): \( s = \int_0^{t_0} (t_0 - t) dt \)
• For \( t > t_0 \): \( s = \int_{t_0}^{t} (t - t_0) dt \)

Calculating these integrals will give you the total distance traveled in each segment. For example:

• For \( t < t_0 \): The integral evaluates to \( s = t_0 t - \frac{t^2}{2} \) evaluated from 0 to \( t_0 \), resulting in \( s = \frac{t_0^2}{2} \).
• For \( t > t_0 \): The integral evaluates to \( s = \frac{t^2}{2} - t_0 t + \frac{t_0^2}{2} \) evaluated from \( t_0 \) to \( t \), leading to a more complex expression depending on the limits.

Real-World Implications

This model can be applied to various real-world scenarios, such as a car accelerating from a stoplight or a vehicle that slows down to a stop before speeding up again. Understanding how to interpret and manipulate such equations is crucial in fields like physics and engineering, where motion analysis is essential.

In summary, the equation \( v = |t - t_0| \) provides a clear framework for understanding the car's velocity over time, and by integrating this function, you can determine the distance traveled during different intervals. This approach not only helps in solving problems but also enhances your grasp of motion dynamics.