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Understanding velocity functions is crucial in kinematics as it gives us an idea of how fast something is moving at any given time. The velocity function represents velocity as a function of time, telling us how speed changes over time. For our race car scenario, the velocity function is given by:

• \( v_x(t) = 0.860 \, t^2 \)
• where \( v_x(t) \) is the velocity at time \( t \), measured in meters per second.

Since this is in the form of \( at^2 \), it suggests that velocity increases with the square of the time, which is typical of uniformly accelerating motion starting from rest (zero initial velocity). This makes determining how fast the car moves at any given instant straightforward by simply plugging in different values of \( t \) into the function.
Understanding this formula sets the stage for deriving the acceleration and solving further complex problems like time calculations.

Acceleration tells us how quickly the velocity is changing over time, and it's fundamental in kinematics for understanding motion. In mathematical terms, acceleration is the rate of change of velocity. For the velocity function \( v_x(t) = 0.860 \, t^2 \), the acceleration \( a(t) \) is the derivative of the velocity function with respect to time:

• Start by differentiating: \( a(t) = \frac{d}{dt}[0.860 \, t^2] = 2 \, \times 0.860 \, t = 1.72 \, t \).

This calculation shows that the acceleration is not constant and depends on time \( t \). Specifically, it indicates a linear increase in acceleration as time progresses. Knowing the acceleration function allows us to find how quickly the car is speeding up at any given point in its journey.

The derivative is a powerful mathematical tool used to understand how quantities change. In this case, it helps us find acceleration from velocity. The derivative of a function provides the slope of the tangent to the function at any point, representing the rate of change of that function. When we took the derivative of \( v_x(t) = 0.860 \, t^2 \), we found the acceleration \( a(t) = 1.72 \, t \).

• This derivative shows how the velocity is increasing over time with a linear relationship.

In essence, the process of taking a derivative allows us to transition from analyzing position to velocity, and from velocity to acceleration, unraveling the complete picture of motion.

Time calculations are an integral part of kinematics, allowing us to determine specific moments during an object's motion. In this exercise, we needed to find out when the velocity \( v_x \) reached 12.0 m/s. By setting \( v_x(t) = 12.0 \) and solving for \( t \), we found the equation:

• \( 0.860 \, t^2 = 12.0 \), leading to \( t^2 = \frac{12.0}{0.860} \)
• Thus, \( t = \sqrt{\frac{12.0}{0.860}} \).

This calculation resulted in \( t \approx 3.73 \) seconds, meaning the car reached that speed at approximately 3.73 seconds. Combining time determinations with velocity and acceleration functions provides insight into the entire motion of an object, making it easier to predict future states or understand past motion.