Problem 2 - Fundamentals of Physics

Question

Compute your averagc velocity in the following two cascs: (a) You walk \(73.2 \mathrm{~m}\) at a speed of \(1.22 \mathrm{~m} / \mathrm{s}\) and then run \(73.2 \mathrm{~m}\) at a speed of \(3.05 \mathrm{~m} / \mathrm{s}\) along a straight track. (b) You walk for \(1.00 \mathrm{~min}\) at a speed of \(1.22 \mathrm{~m} / \mathrm{s}\) and then run for \(1.00 \mathrm{~min}\) at \(3.05 \mathrm{~m} / \mathrm{s}\) along a straight track. (c) Graph \(x\) versus \(t\) for both cases and indicate how the average velocity is found on the graph.

Step-by-Step Solution

Answer

In case (a), average velocity is determined by total time and distance covered. In case (b), it's found similarly, but over equal times. The average velocity is represented graphically as the slope of the line connecting the start and endpoint.

Step 1: Analyze Case (a) Walking and Running at Different Speeds

In this case, you cover different distances at different speeds. First, calculate the time taken to walk and run. The distance walked is given as 73.2 m at a speed of 1.22 m/s. Therefore, the time taken to walk is:\[ t_1 = \frac{73.2 \text{ m}}{1.22 \text{ m/s}} \]Calculate this value, then do the same for running:The distance run is the same 73.2 m at a speed of 3.05 m/s. Therefore, the time taken is:\[ t_2 = \frac{73.2 \text{ m}}{3.05 \text{ m/s}} \]

Step 2: Compute Total Time and Total Distance for Case (a)

Add up the times from walking and running to find the total time, then compute the total distance covered:\[ t_\text{total} = t_1 + t_2 \]The total distance is simply:\[ x_\text{total} = 73.2 \text{ m} + 73.2 \text{ m} = 146.4 \text{ m} \]

Step 3: Calculate Average Velocity for Case (a)

Average velocity is defined as total distance divided by total time:\[ v_\text{avg} = \frac{x_\text{total}}{t_\text{total}} \]Insert the total distance and total time from the previous steps to calculate the average velocity for Case (a).

Step 4: Analyze Case (b) Walking and Running for Equal Durations

In this case, the time spent walking and running is given, so we need to calculate the distance covered walking and running.For walking, with a speed of 1.22 m/s for 60 seconds (1 minute), the distance is:\[ x_1 = 1.22 \text{ m/s} \times 60 \text{ s} \]For running, with a speed of 3.05 m/s for 60 seconds, the distance is:\[ x_2 = 3.05 \text{ m/s} \times 60 \text{ s} \]

Step 5: Compute Total Distance and Total Time for Case (b)

The total distance is the sum of the distances walked and run:\[ x_\text{total} = x_1 + x_2 \]The total time is:\[ t_\text{total} = 60 \text{ s} + 60 \text{ s} = 120 \text{ s} \]

Step 6: Calculate Average Velocity for Case (b)

Now, use the definition of average velocity for Case (b):\[ v_\text{avg} = \frac{x_\text{total}}{t_\text{total}} \]Insert the total distance and total time to find the average velocity for Case (b).

Step 7: Graphical Representation for Both Cases

Graph the position \(x\) versus time \(t\) for both scenarios.For case (a), plot two segments: the walking segment with a gentler slope, and the running segment with a steeper slope.For case (b), plot two segments of equal time: the first, a gentler slope for walking, and the second, a steeper slope for running.Indicate that the average velocity is the overall slope of the line connecting the start and end points on these graphs.

Key Concepts

Kinematics

Kinematics is a branch of mechanics that deals with the motion of objects without focusing on the forces causing the motion. Imagine watching a car move along a track and simply observing how fast and where it goes over time. This is what kinematics is all about. It helps us describe the motion of an object using quantities like distance, speed, velocity, and time.

In kinematics, velocity is an important concept. Velocity tells us how quickly an object is moving and in what direction. Unlike speed, which is only about how fast something is moving, velocity includes direction too. For example, the velocity of a car traveling east at 60 km/h is different from a car traveling west at the same speed. Average velocity is calculated by dividing the total displacement (change in position) by total time. This gives us a simple way to describe the object's overall motion.

When working with kinematics problems, keep in mind that:

• Consistent units are crucial – for instance, if using meters and seconds, stick with them throughout.
• Displacement is different from distance – displacement considers the initial and final positions, while distance is the total path traveled.
• Velocity can be positive or negative – depending on the chosen direction of motion.

Average Speed

Average speed is a measure of how much distance an object covers over a period of time, regardless of direction. It's a scalar quantity, which means it doesn't take direction into account, unlike velocity, which does.

To find the average speed, you divide the total distance traveled by the total time taken. Here's the formula for average speed:\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]For example, if you walk 50 meters and then run another 50 meters, covering a total of 100 meters in 25 seconds, your average speed would be:\[ \text{Average Speed} = \frac{100 \text{ m}}{25 \text{ s}} = 4 \text{ m/s} \]It's important to note that average speed doesn't reveal much about the journey’s specifics – only how much ground was covered over a certain time. It doesn’t tell you where you started or finished, just the total distance traversed.

When working on problems involving average speed, remember:

• Total distance considers the entire path traveled, not just point-to-point displacement.
• Time should encompass the entire journey, from start to finish.
• It's purely a measure of how much ground was covered and thus ignores travel direction.

Distance-Time Graph

A distance-time graph visually represents an object's journey, showing how distance varies with time. It's a valuable tool in kinematics because it allows us to quickly interpret data about motion, such as speed and distance traveled.

On this graph:

• The horizontal axis (x-axis) represents time.
• The vertical axis (y-axis) represents distance.

The slope of the line on a distance-time graph gives us the speed of the object. A steeper slope indicates the object is moving faster, while a shallower slope suggests slower movement. In our exercise, when you're walking, the slope is gentler, and when you're running, it's steeper.

To find the average velocity from a distance-time graph, you draw a line from the starting point to the end point of the motion. The slope of this line is the average velocity. For example, in case exercises:

• In Case (a), where different speeds are used, the graph has segments of varying slopes.
• In Case (b), where equal durations are given for walking and running, segments have equal width on the time axis but varying slopes.

By analyzing a distance-time graph, you can gain insights not only into the total distance traveled and speed but also patterns in motion over the period. For instance, any curved sections indicate acceleration or deceleration. Understanding a distance-time graph helps to grasp how movement changes over time, a key part of kinematics.