Problem 16 - Applied Calculus

Question

At time \(t\) in seconds, a particle's distance \(s(t),\) in \(\mathrm{cm}\) from a point is given in the table. What is the average velocity of the particle from \(t=3\) to \(t=10 ?\) $$\begin{array}{c|c|c|c|c|c} \hline t & 0 & 3 & 6 & 10 & 13 \\ \hline s(t) & 0 & 72 & 92 & 144 & 180 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The average velocity is approximately 10.29 cm/s.

Step 1: Identify Time Interval

The average velocity is calculated over a specific time interval. Here, we are interested in the interval from \( t = 3 \) to \( t = 10 \).

Step 2: Determine Position at Start and End

From the table, determine the position of the particle at the start and end of the time interval. At \( t = 3 \), the position \( s(3) \) is 72 cm. At \( t = 10 \), the position \( s(10) \) is 144 cm.

Step 3: Compute Change in Position

The change in position \( \Delta s \) over the interval is computed by subtracting the initial position from the final position: \[ \Delta s = s(10) - s(3) = 144 - 72 = 72 \text{ cm} \].

Step 4: Compute Time Interval Duration

Calculate the duration of the time interval: \[ \Delta t = 10 - 3 = 7 \text{ seconds} \].

Step 5: Calculate Average Velocity

Average velocity is defined as the change in position divided by the change in time. Thus, compute it using the formula: \[ \text{Average velocity} = \frac{\Delta s}{\Delta t} = \frac{72}{7} \approx 10.29 \text{ cm/s} \].

Key Concepts

Understanding Kinematics

Kinematics is a branch of physics that focuses on the motion of objects without considering the forces that cause this motion. It is primarily concerned with quantities like displacement, velocity, and acceleration.
In the context of our problem, kinematics helps us to understand how the particle's motion is described using these quantities. We observe the particle's position at various times to determine its velocity over specific intervals.

• Displacement: The overall change in position of the particle.
• Velocity: The rate of change of displacement concerning time.
• Acceleration: The rate of change of velocity over time.

The focus here is on average velocity, which gives us a sense of the particle's overall speed and direction over a specified time interval.

Basics of Distance Measurement

In the context of kinematics, distance is the measure of how much ground an object covers during its motion. However, in problems like ours, we often use a concept known as displacement, which is different from distance because it considers only the initial and final positions.
Displacement is a vector quantity, meaning it has both magnitude and direction, allowing us to calculate exact position shifts. For example, in our exercise, the particle starts at 72 cm and ends at 144 cm within the chosen interval, giving us a displacement of 72 cm.
The reason displacement is used more prominently in kinematics is due to its ability to succinctly describe an object's change in position, vital for calculating velocities and other motion parameters.

Importance of Time Interval

The time interval is a crucial aspect of analyzing motion in kinematics. It denotes the duration over which observations regarding position changes are made.
In the given exercise, we compute the average velocity over the specified time interval from 3 to 10 seconds.

• The time interval is calculated simply by subtracting the initial time from the final time.
• We determined this to be 7 seconds in our example.

This interval plays a critical role in calculating the velocity since velocity is always described as a function over time. Understanding this concept ensures that students can apply similar calculations across various scenarios, regardless of the complexity of object motion.

Concept of Particle Motion

Particle motion refers to how a particle moves through space over time. In our exercise, the particle has its position marked at different times, which allows us to understand its movement pattern.
The motion can be described in terms of displacement and velocity, as seen in the given exercise. By looking at different time points, we can see how the particle accelerates or decelerates, which is crucial in more advanced studies.
In simple terms, particle motion helps us to visualize and calculate how, where, and at what speed a particle travels, which is fundamental in many practical applications, from engineering to animation.