Problem 23 - 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 of the particle from \(t=3\) to \(t=10\) is approximately 10.29 cm/s.

Step 1: Understand the Formula for Average Velocity

The average velocity of an object over a time interval is defined as the change in displacement divided by the change in time. The formula is: \[ \text{Average velocity} = \frac{\Delta s}{\Delta t} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \] where \(s(t_2)\) and \(s(t_1)\) are the distances at times \(t_2\) and \(t_1\), respectively.

Step 2: Identify Given Values

From the table, identify the values of \(s(t_1)\) and \(s(t_2)\), where \(t_1 = 3\) and \(t_2 = 10\). From the table, \(s(3) = 72\) and \(s(10) = 144\). The time interval is from \(t_1 = 3\) to \(t_2 = 10\).

Step 3: Calculate Change in Displacement

Substitute \(s(3)\) and \(s(10)\) into the formula to calculate the change in displacement:\[ \Delta s = s(10) - s(3) = 144 - 72 = 72 \text{ cm} \]

Step 4: Calculate Change in Time

Calculate the change in time:\[ \Delta t = t_2 - t_1 = 10 - 3 = 7 \text{ seconds} \]

Step 5: Compute Average Velocity

Substitute \(\Delta s = 72\) cm and \(\Delta t = 7\) seconds into the average velocity formula:\[ \text{Average velocity} = \frac{72}{7} \approx 10.29 \text{ cm/s} \]

Key Concepts

Displacement

Displacement refers to an object's change in position, measured from a starting point to an ending point. It provides a straight line measure, regardless of the path traveled in between. To calculate displacement, simply subtract the initial position from the final position. In the example provided, you find that at time \(t_1 = 3\), the position \(s(3) = 72\) cm, and at time \(t_2 = 10\), the position \(s(10) = 144\) cm.
\[ \Delta s = s(10) - s(3) = 144 \text{ cm} - 72 \text{ cm} = 72 \text{ cm} \]
This change in position is the displacement over the given time interval, showing how far the particle moved in a straight line.

Time Interval

The time interval represents the duration over which an event occurs, or in this context, it is the span between two points in time during which the displacement happens. Calculating a time interval is done by subtracting the initial time from the final time. Using the values given, when \(t_1 = 3\) seconds and \(t_2 = 10\) seconds, the time interval becomes:
\[ \Delta t = t_2 - t_1 = 10 - 3 = 7 \text{ seconds} \]
This time interval is essential for determining the average velocity, as it indicates over what period the displacement took place.

Velocity Formula

The velocity formula plays a crucial role in determining how fast an object moves in a particular direction over a given time. It is defined as the change in displacement divided by the change in time. The formula used for average velocity is:
\[ \text{Average velocity} = \frac{\Delta s}{\Delta t} \]
For the particle, given \(\Delta s = 72 \, \text{cm}\) and \(\Delta t = 7 \, \text{seconds}\), the average velocity is:
\[ \text{Average velocity} = \frac{72}{7} \approx 10.29 \, \text{cm/s} \]
This formula is very useful as it provides the average speed of the particle over the specified time interval, giving a snapshot of the motion.