Problem 1 - Calculus

Question

The distance, \(s,\) a car has traveled on a trip is shown in the table as a function of the time, \(t,\) since the trip started. Find the average velocity between \(t=2\) and \(t=5\) $$\begin{array}{c|c|c|c|c|c|c}\hline t \text { (hours) } & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline s(\mathrm{km}) & 0 & 45 & 135 & 220 & 300 & 400 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The average velocity is approximately 88.33 km/hr.

Step 1: Identify Initial and Final Points

From the table, we find the distance at two specific times. At time \( t = 2 \) hours, the distance \( s = 135 \) km, and at time \( t = 5 \) hours, the distance \( s = 400 \) km.

Step 2: Calculate Change in Distance

Subtract the initial distance from the final distance to find the change in distance. The change in distance is \( 400 - 135 = 265 \) km.

Step 3: Calculate Change in Time

Subtract the initial time from the final time to find the change in time. The change in time is \( 5 - 2 = 3 \) hours.

Step 4: Compute Average Velocity

The average velocity is calculated by dividing the change in distance by the change in time. Average velocity \( v_{avg} = \frac{265 \text{ km}}{3 \text{ hr}} \approx 88.33 \text{ km/hr} \).

Key Concepts

Change in Distance

Change in distance refers to how much a car, object, or person moves over a certain period. In this exercise, it's the difference between distances at two times. For example, the car first travels 135 km at 2 hours into the trip. Finally, it reaches 400 km at 5 hours. To find how much distance changed, subtract the distance at the starting point (135 km) from the final distance (400 km). This gives us a change in distance of 265 km. Understanding change in distance helps grasp how movement differs over time. It connects to the wider concept of motion in physics, which is required to determine things like speed and velocity. By recognizing this change, students learn to calculate speed and also predict future positions.

Change in Time

Change in time is the duration or the difference between two time points when observing movement or change. In this case, it helps us find how long the car took to cover a distance. At first, the car is at 2 hours, and it moves to the point where 5 hours have elapsed. Subtracting the initial time (2 hours) from the final time (5 hours) shows a change of 3 hours. Understanding change in time provides clarity about how long an event or action happens. Knowing the time taken to cover a certain distance helps in calculating other quantities like speed. It also supports critical thinking by enabling problem-solving in timed scenarios.

Distance-Time Table

A distance-time table visually and systematically represents how distance changes over time. It is an essential tool, especially when calculating velocity. The table in this exercise gives the distance (in km) at various times (in hours). Each row corresponds to a different point in time and the corresponding distance. This specific table lets us quickly read off critical information needed to calculate average speed and other related quantities. Using a distance-time table is effective for tracking movement:

• Students can identify starting and stopping points easily.
• It allows quick comparison between different time periods.
• It displays the trend of motion over time.

These tables make complex data manageable, leading to better insights about how objects behave over time.