Problem 49 - Calculus for Biology and Medicine

Question

A car moves in a straight line. At time \(t\) (measured in seconds), its position (measured in meters) is $$ s(t)=\frac{1}{100} t^{3}, 0 \leq t \leq 5 $$ (a) Find its average velocity between \(t=0\) and \(t=5\). (b) Find its instantaneous velocity for \(t \in(0,5)\). (c) At what time is the instantaneous velocity of the car equal to its average velocity?

Step-by-Step Solution

Answer

(a) The average velocity is 0.25 m/s. (b) Instantaneous velocity is \( \frac{3}{100}t^2 \). (c) Instantaneous equals average at approximately 2.887 seconds.

Step 1: Understand Average Velocity

Average velocity is calculated by the change in position over the change in time. It can be expressed with the formula: \( v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \). In this case, we are given \( s(t) = \frac{1}{100}t^3 \), and we need to find \( v_{avg} \) between \( t = 0 \) and \( t = 5 \).

Step 2: Calculate Average Position Change

First, find \( s(t) \) at the given times.Calculate \( s(0) = \frac{1}{100} \times 0^3 = 0 \) and \( s(5) = \frac{1}{100} \times 5^3 = \frac{125}{100} = 1.25 \text{ meters} \).

Step 3: Compute Average Velocity

Use the average velocity formula: \( v_{avg} = \frac{s(5) - s(0)}{5 - 0} = \frac{1.25 - 0}{5} = 0.25 \text{ meters per second}\).

Step 4: Find Instantaneous Velocity Formula

Instantaneous velocity is the derivative of the position function \( s(t) \) with respect to \( t \). Differentiate \( s(t) = \frac{1}{100}t^3 \) to get \( v(t) = \frac{d}{dt}\left(\frac{1}{100}t^3\right) = \frac{3}{100}t^2 \).

Step 5: Solve for Instantaneous Velocity Equal to Average

Set the instantaneous velocity \( v(t) = \frac{3}{100}t^2 \) equal to the previously calculated average velocity \( 0.25 \):\( \frac{3}{100}t^2 = 0.25 \rightarrow 3t^2 = 25 \rightarrow t^2 = \frac{25}{3} \rightarrow t = \sqrt{\frac{25}{3}} \approx 2.887 \text{ seconds} \).

Key Concepts

Average Velocity

Average velocity is a fundamental concept in calculus, especially in fields like biology and medicine where motion and change are often analyzed. It represents the total change in position divided by the total time spent to make that change. This gives us an overall picture of how fast an object was moving between two points in time.

When we look at the position function given by \(s(t) = \frac{1}{100}t^3\), average velocity helps us understand the general movement of the car from \(t = 0\) to \(t = 5\) seconds.

To calculate average velocity, we use the formula:

• \(v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}\)

For this exercise, \(t_1 = 0\) and \(t_2 = 5\). You plug these values into the position function to get:

• \(s(0) = \frac{1}{100} \times 0^3 = 0 \text{ meters}\)
• \(s(5) = \frac{1}{100} \times 5^3 = 1.25 \text{ meters}\)

Then, calculate the average velocity:

• \(v_{avg} = \frac{1.25 - 0}{5 - 0} = 0.25 \text{ meters per second}\)

This means on average, the car traveled at 0.25 meters per second over the 5 seconds.
Understanding average velocity is crucial when dealing with non-uniform motion in biology such as population migration or blood flow rates.

Instantaneous Velocity

Instantaneous velocity gives a snapshot of how fast an object is moving at any given moment. Unlike average velocity, which smooths out changes in speed over time, instantaneous velocity shows us the exact speed at every precise second.

In calculus, this is found by taking the derivative of the position function with respect to time. The derivative provides us with a new function, representing the rate of change of position or velocity.

For the function \(s(t) = \frac{1}{100}t^3\), the instantaneous velocity is derived as follows:

• Calculate the derivative: \(v(t) = \frac{d}{dt}\left(\frac{1}{100}t^3\right) = \frac{3}{100}t^2\)

This result, \(v(t) = \frac{3}{100}t^2\), tells us the speed of the car at any time \(t\).
It is a dynamic piece of information crucial when examining rapid biological processes, such as enzyme reactions or neural signaling.

Instantaneous velocity is key in situations where speed changes frequently over short periods.

Position Function

The position function is at the heart of analyzing motion in calculus. It provides information on where an object is at any given time. Understanding this function is crucial for both average and instantaneous velocity calculations, as it establishes the foundation from which these velocities are derived.

In the given exercise, the position function is \(s(t) = \frac{1}{100}t^3\). This cubic function indicates how the car's position changes over time:

• For \(t = 0\), the position is 0 meters.
• For \(t = 5\), the position is 1.25 meters as previously calculated.

The position function enables us to observe not only the initial and final positions but every point in between.

Understanding this concept helps in fields like biology, where analyzing trajectories - be it the movement of cells, viruses, or animals - is often necessary.By using the position function, we facilitate the analysis of motion and change, which are essential for modeling biological processes.