Question

Show that if an object's position function is given by \(s(t)=a t^{2}+b t+c\), then the average velocity over the interval \([A, B]\) is equal to the instantaneous velocity at the midpoint of \([A, B] .\)

Short answer

The average velocity equals the instantaneous velocity at the midpoint: \(a(A+B) + b\).

Step-by-step solution

Find the Average Velocity Formula

The formula for average velocity over an interval \[A, B\] is determined by the change in position over the change in time. Thus, the average velocity \(v_{avg}\) is given by: \[ v_{avg} = \frac{s(B) - s(A)}{B - A} \] Substitute the expression for \(s(t) = at^2 + bt + c\) to get: \[ v_{avg} = \frac{aB^2 + bB + c - (aA^2 + bA + c)}{B-A} \] Simplifying further we get: \[ v_{avg} = \frac{a(B^2 - A^2) + b(B - A)}{B-A} \] Using the difference of squares \(B^2 - A^2 = (B-A)(B+A)\), the expression simplifies to: \[ v_{avg} = a(B+A) + b \]

Determine the Instantaneous Velocity

The instantaneous velocity \(v_{inst}\) is the derivative of the position function \(s(t)\). Therefore, \[ v(t) = \frac{d}{dt}(at^2 + bt + c) = 2at + b \] To find the instantaneous velocity at the midpoint of the interval \([A, B]\), calculate the midpoint \(M = \frac{A + B}{2}\). Substitute \(M\) into \(v(t)\): \[ v_{inst}(M) = 2a \left(\frac{A + B}{2}\right) + b = a(A + B) + b \]

Compare Average and Instantaneous Velocity

From Steps 1 and 2, we have: - Average velocity: \(v_{avg} = a(B+A) + b\)- Instantaneous velocity at the midpoint: \(v_{inst}(M) = a(A + B) + b\) Both expressions are equal, thus proving the required equality.

Key concepts

Position Function

In the realm of physics and calculus, the position function is a critical concept. It describes an object's location concerning time. In mathematical terms, if we have a position function represented as \( s(t) = at^2 + bt + c \), this function tells us where an object is along a straight line at any given time \( t \). - **Components**: - The term \( at^2 \) represents the influence of acceleration on the position. - The term \( bt \) accounts for the initial velocity of the object. - The constant \( c \) is the starting position of the object when time \( t = 0 \). The position function not only tells us the current position but also provides the foundation for calculating other aspects like velocity.

Instantaneous Velocity

Instantaneous velocity is a concept you come across when you want to know how fast something is moving at exactly one moment in time. Unlike average velocity, which looks at motion over a period, instantaneous velocity gives you the speed at a specific second.- **Derivative Connection**: - To find instantaneous velocity, you'll need to take the derivative of the position function. - For our position function \( s(t) = at^2 + bt + c \), the derivative is \( v(t) = 2at + b \). This derivative tells you how steep the position line is at any point \( t \), which corresponds to how fast the object is going right then. The formula \( v(t) = 2at + b \) is essential whenever you wish to capture the dynamic nature of an object's speed on a precise scale.

Differentiation

Differentiation is a mathematical process applied to functions to find the rate at which something changes. In simple terms, it's about figuring out how a curve (like our position function) behaves at any given point.- **Application**: - For the position function \( s(t) = at^2 + bt + c \), differentiation helps us determine the instantaneous velocity, as mentioned. - By differentiating the function, we get the velocity function, \( v(t) = 2at + b \), showing how position changes with time.Differentiation is powerful for analyzing movements because it breaks down how things shift and transforms to a moment-to-moment understanding. In various fields like physics and engineering, it's a go-to method for finding things like speed and acceleration.

Midpoint of Interval

The midpoint of an interval in mathematics is a simple yet useful concept. It refers to the point that is exactly halfway between two numbers \( A \) and \( B \) on a number line.- **Calculation**: - The midpoint \( M \) is calculated using the formula \( M = \frac{A + B}{2} \). - This average of the two endpoints provides a central point to measure other values, like velocity in our problem.Understanding the midpoint is crucial because it provides a fair representation of an interval. In our exercise, the instantaneous velocity at this midpoint was found to equal the average velocity over the entire interval, a profound insight that demonstrates the consistency in the object's motion.