Question

For each of the following quantities, indicate whether it is a scalar or a vector: (a) the time it takes you to run the \(100-m\) dash, (b) your displacement after running the \(100-\mathrm{m}\) dash, (c) your average velocity while running, (d) your average speed while running.

Short answer

(a) Scalar, (b) Vector, (c) Vector, (d) Scalar.

Step-by-step solution

Understanding Scalars and Vectors

In physics, a scalar is a quantity that has only magnitude, such as time and speed. A vector has both magnitude and direction, such as displacement and velocity.

Analyzing Quantity (a)

The time it takes you to run the 100-m dash is a measurement of duration. Since time is a quantity that has only magnitude without any directional component, it is a scalar.

Analyzing Quantity (b)

Displacement refers to the change in position and includes a direction component because it occurs in a specific direction from the starting point. Therefore, displacement is a vector.

Analyzing Quantity (c)

Average velocity includes both the magnitude (how fast something is moving) and the direction (the specific path of movement). Thus, it is classified as a vector.

Analyzing Quantity (d)

Average speed is the total distance traveled divided by the time taken. It has only magnitude and lacks a direction element, making it a scalar.

Key concepts

Displacement

Displacement is a fundamental concept in physics that describes the change in position of an object. It's important to understand that displacement is not simply the distance an object travels. Instead, it refers to the shortest path between the initial and final positions, taking direction into account. Since displacement includes both magnitude (the size of the change in position) and direction (the way it's headed), it is classified as a vector quantity.
For example, if you run in a straight line from point A to point B, your displacement will be the direct line connecting these points, including the direction. If you return back to point A after reaching point B, your total displacement is zero, even if you've covered a specific distance.

Average Velocity

Average velocity is a vector quantity that gives us information about an object's speed and direction over a period. It's calculated by dividing the total displacement by the total time taken. The formula is:\[\text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}}.\]
This means that average velocity provides a full picture of how fast and where an object is moving. For instance, if you walked 10 meters east and reached the destination in 2 seconds, your average velocity would be 5 meters per second eastward. Keep in mind that since it accounts for direction, changes in direction affect it significantly. If you moved back to your starting point, resulting in zero displacement, your average velocity would also be zero.

Average Speed

Average speed, unlike average velocity, is a scalar quantity. This means it only considers the magnitude of motion, leaving out the direction component. Average speed is calculated by dividing the total distance traveled by the total time taken:\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}.\]
A key point here is that average speed doesn't change if the direction changes. For instance, if you ran around a track and returned to your start point, you'd have traveled a particular distance, resulting in a measurable average speed. This measurement does not become zero like average velocity would, because it ignores the direction of movement. Average speed always gives a positive value, offering a simple way to gauge how fast you're moving regardless of direction.

Magnitude and Direction

Magnitude and direction are terms often used when dealing with vectors. The magnitude is the size or length of the vector, while the direction gives the orientation. Understanding both is crucial when analyzing vector quantities.

• Magnitude refers to how much of something there is. For distance, it's in units like meters or kilometers. In terms of force, it's newtons or pounds.
• Direction indicates the line along which something moves or faces. This could be specified in degrees from a point, using compass points or as positive or negative directions on a coordinate axis.

When you break down a vector like displacement or velocity, you see that the magnitude tells "how much" and the direction tells "which way." Both these aspects are needed to fully describe a vector quantity. Without both, the information would be incomplete and potentially misleading in physical terms.