Question

Analyze Friends tell you that on a recent trip their average velocity was \(+20 \mathrm{~m} / \mathrm{s}\). Is it possible that at any time during the trip their velocity was \(-20 \mathrm{~m} / \mathrm{s}\) ? Explain, and give an example to justify your answer.

Short answer

Yes, they could have had a velocity of \(-20 \mathrm{~m/s}\) if they reversed direction during the trip.

Step-by-step solution

Understanding Average Velocity

The average velocity is calculated as the total displacement divided by the total time taken. Mathematically, it can be represented as \( v_{avg} = \frac{\Delta x}{\Delta t} \), where \( \Delta x \) is the total displacement and \( \Delta t \) is the total time.

Analyzing the Value of Average Velocity

Given that the average velocity is \(+20 \mathrm{~m/s}\), it indicates that the overall or net change in position was in the positive direction. However, this does not directly inform us about the velocities during the trip at specific moments.

Considering Instantaneous Velocity

An instantaneous velocity can differ from the average velocity. During the trip, there might be instances where different velocities, even negative ones, occur, provided these don't change the overall displacement such that the average becomes negative.

Possibility of Negative Instantaneous Velocity

Since velocity is a vector quantity (having both magnitude and direction), it’s possible to momentarily have \(-20 \mathrm{~m/s}\), for example when changing direction, however briefly, as long as the net displacement over time remains in the positive direction to keep the average at \(+20 \mathrm{~m/s}\).

Example Scenario

Imagine a scenario where a vehicle starts moving forward (positive direction) at \(+40 \mathrm{~m/s}\) for the first half of the journey's duration and reverses (negative direction) at \(-20 \mathrm{~m/s}\) for the second half. Over time, the average velocity can still compute to \(+20 \mathrm{~m/s}\) depending on distances covered and reversing time.

Key concepts

Instantaneous Velocity

Instantaneous velocity refers to the speed and direction of an object at a particular moment in time. Imagine it as capturing a snapshot of a vehicle's speedometer at an exact instant during a trip. Unlike average velocity, which is spread over the entire duration of a journey, instantaneous velocity can vary significantly at different points.
For instance, if you glance at the speedometer when your car ascends a steep hill, you might find it momentarily slower than when descending. This is an example of how instantaneous velocity can differ constantly, reflecting every little change in motion.

• Instantaneous velocity provides a detailed look into motion.
• It is often expressed in the form of a derivative, for instance, \( v(t) = \frac{dx}{dt} \), highlighting how position changes with time at any given point.
• This aspect allows for negative values, capturing changes in direction.

Thus, even if your average velocity is positive, the instantaneous velocity at any particular moment can indeed be negative during direction changes, taking you momentarily backward.

Displacement

Displacement is all about the net change in position of an object from start to end point, regardless of the path taken between the two. It's crucial because it differs from the total distance traveled. Imagine walking 5 meters north and then 5 meters south. Your total distance is 10 meters, but your displacement is zero because you end up back where you started.
In the context of your friends' trip, their average velocity being \(+20 \text{ m/s}\) indicates a net positive displacement. They moved further in one direction compared to any backtracking. Displacement is crucial for calculating average velocity because it includes both magnitude and direction.

• Defined vectorially as \( \Delta x = x_{final} - x_{initial} \).
• Displacement is a straight-line measurement, making it different from mere distance.
• This vector nature makes it indispensable in finding average velocities.

It's the directionality of displacement that helps interpret motion complexities, like experiencing forward and backward travel.

Vector Quantity

Vector quantities are fundamental in physics because they possess both magnitude and direction. Velocity is such a quantity, distinguishing it from speed, which only describes magnitude. Understanding vector quantities aids in explaining how different motion segments contribute to overall outcomes, like average velocity.
Consider your friends on their trip: even though they had an average velocity of \(+20 \text{ m/s}\), they could've also experienced velocities in opposing directions. With vectors, the positive direction could represent moving towards a destination, and any negative velocity signifies moving away from it.

• Vectors are represented by arrows in diagrams, depicting both size and direction.
• Common operations with vectors include addition, subtraction, and scalar multiplication, all following specific rules.
• Aligned with vector math, analyzing velocities involves calculations over different intervals or directions.

This vectorial approach validates how different directional velocities can coexist, producing a net result as observed in average velocities.

Direction of Motion

The direction of motion is crucial in determining how velocity vectors influence overall movement. It's the compass of motion that can completely alter how we perceive velocity.
For your friend's trip, having an instantaneous velocity of \(-20 \text{ m/s}\) suggests movement contrary to the positive path direction used in the average velocity calculation. This means briefly moving in the opposite direction.

• Movement direction is essential for understanding velocity vectors.
• Having varied movement directions during a trip can result in an average velocity that might not state the entire motion story.
• Evaluating direction allows for the insight that backward motion (negative), can occur without affecting the positive average travel direction.

Recognizing these directional changes underpins our comprehension of velocity calculations, distinguishing between mere speed and true vectorial motion dynamics.