Question

Compute your average speed in the following two cases. ( \(a\) ) You walk \(240 \mathrm{ft}\) at a speed of \(4.0 \mathrm{ft} / \mathrm{s}\) and then run \(240 \mathrm{ft}\) at a speed of \(10 \mathrm{ft} / \mathrm{s}\) along a straight track. ( \(b\) ) You walk for \(1.0\) min at a speed of \(4.0 \mathrm{ft} / \mathrm{s}\) and then run for \(1.0 \mathrm{~min}\) at \(10 \mathrm{ft} / \mathrm{s}\) along a straight track.

Short answer

The average speed in case (a) is 5.71 ft/s, and in case (b) it is 7.0 ft/s.

Step-by-step solution

Determine total distance for each scenario

For both cases (a) and (b), the total distance would be the sum of the distances covered during the walking phase and the running phase. In case (a), the total distance is \(240 + 240 = 480\) ft. In case (b), since time and speed are known, we can calculate the distance covered using the formula distance = speed × time. Thus, distance covered during the walk is \(4 * 60 = 240\) ft, and distance covered during run is \(10 * 60 = 600\) ft. Summing these values gives us a total distance of \(240 + 600 = 840\) ft.

Calculate total time for each scenario

In case (a), since distance and speed are given, we can calculate time using the formula time = distance / speed. Thus, time for the walk is \(240 / 4 = 60\) s and time for the run is \(240 / 10 = 24\) s. Add these values to get the total time: \(60 + 24 = 84\) s. In case (b), the total time of the walk and the run is given to be \(1.0 + 1.0 = 2.0\) mins or \(120\) s.

Compute the average speed for each scenario

Now we can calculate the average speed by using the formula average speed = total distance / total time. For case (a), the average speed is \(480 / 84 = 5.71\) ft/s. For case (b), the average speed is \(840 / 120 = 7.0\) ft/s.

Key concepts

Distance Formula in Physics

Understanding the distance formula in physics is crucial for accurately calculating movement in space. In essence, the distance formula relates the amount of space covered during motion to the speed at which an object travels and the time it takes to cover that space. It is effectively summarized by the equation:
\[ \text{distance} = \text{speed} \times \text{time} \]
This relationship is foundational when dealing with problems in mechanics and kinematics. For instance, if a person walks at a steady pace of 4 feet per second (ft/s) for 1 minute, we can determine the distance they've walked by multiplying their speed by the time in seconds (since 1 minute equals 60 seconds).
\[ \text{distance} = 4\,\text{ft/s} \times 60\,\text{s} = 240\,\text{ft} \]
However, in real-world situations, speeds can vary, and so the total distance covered might involve calculations for different speeds over different periods. It's essential to consider all parts of a journey to find the overall distance traveled.

Time-Speed Relationship

The relationship between time and speed is a fundamental aspect of many physical situations. Speed is defined as the distance covered per unit of time. The time-speed relationship is especially important when the speed varies over the course of a trip. For calculating the amount of time spent, we use:
\[ \text{time} = \frac{\text{distance}}{\text{speed}} \]
This formula allows us to determine how long an object has been moving at a specific speed to cover a certain distance. For example, if someone runs 240 feet at a rate of 10 feet per second, the time they take is given by the formula:
\[ \text{time} = \frac{240\,\text{ft}}{10\,\text{ft/s}} = 24\,\text{s} \]
Understanding this relationship is not just important in theoretical physics but also in practical situations like sports, driving, and even in daily activities.

Computing Average Speed

Computing the average speed of an object is key to understanding its overall performance over a distance. Average speed is not simply the mean of various speeds but rather the total distance traveled divided by the total time taken:
\[ \text{average speed} = \frac{\text{total distance}}{\text{total time}} \]
When computing average speed, one must account for all the intervals of a trip, including any variations in speed. For instance, having walked a distance at one speed and then run the same distance at a higher speed, as given in our exercise example, requires a careful calculation which looks at the sum of the distances divided by the sum of the individual times spent walking and running.
For variable speeds, such as in the case where a person walks for a minute and then runs for another, adding both distances and dividing by the total time will yield their average speed for the journey:
\[ \text{average speed} = \frac{240\,\text{ft} + 600\,\text{ft}}{120\,\text{s}} = 7\,\text{ft/s} \]

• An understanding of average speed is beneficial for planning travel times.
• It is also used in physics to understand the energetics of moving bodies.
• In daily life, it is universally applied to everything from setting speed limits to estimating travel times.