Question

Check For each of the following quantities, indicate whether it is a scalar or a vector: (a) the time it takes you to run the \(100-\mathrm{m}\) 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

Identify Scalars and Vectors

First, let's clarify the two types: **Scalars** are quantities with just magnitude (size or number) but no direction. Examples include time, speed, area, volume, etc. **Vectors** are quantities that have both magnitude and direction, such as displacement, velocity, force, etc.

Analyze the Time Quantity (a)

The time it takes you to run the 100-m dash is a scalar quantity. Time has a magnitude (for example, 12 seconds) but no direction.

Analyze the Displacement Quantity (b)

Displacement refers to how far and in what direction the runner ends up from the starting point. It is a vector quantity because it includes both the magnitude (100 m, for example) and a direction (say, towards East).

Analyze the Average Velocity Quantity (c)

Average velocity is a vector quantity. It includes information about both the speed of your movement and the direction in which you moved (e.g., 10 m/s East).

Analyze the Average Speed Quantity (d)

Average speed, unlike velocity, is a scalar quantity. It only represents how fast you are moving regardless of the direction (e.g., 10 m/s).

Key concepts

Displacement

Displacement is an important concept in physics when understanding motion. It refers to the change in position of an object. Think of it as the shortest path from the starting point to the final point. Unlike distance, which is a scalar and considers only magnitude, displacement is a vector quantity. This means it possesses both a magnitude and a direction.
For example, if you start at one end of a football field and move to the other end, your displacement might be 100 meters north. Here, '100 meters' indicates the magnitude, and 'north' specifies the direction. This makes displacement a more informative measure than distance when examining the motion of objects.

Average Velocity

Average velocity is crucial for understanding how swiftly and in what direction an object travels over a period. It is calculated by dividing the displacement by the time taken. As a vector quantity, it provides a full description of motion, unlike average speed.
The formula for average velocity is:\[\text{Average velocity} = \frac{\text{Displacement}}{\text{Time}}\]For instance, if a runner covers a displacement of 100 meters east in 10 seconds, the average velocity is 10 meters per second east. The direction east is essential for defining the motion clearly. Hence, average velocity can help determine not only how fast something is moving but in which direction this movement occurs.

Average Speed

Average speed is a measure of how fast an object moves over a certain distance. It differs from velocity as it does not consider the direction. Being a scalar quantity, average speed simply involves magnitude.
The formula for average speed is:\[\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}\]Suppose a runner completes a 100-meter dash in 10 seconds. The average speed would then be 10 meters per second. While it gives an idea of the runner's quickness, it doesn't tell us the direction in which the runner traveled. Therefore, average speed is straightforward but less detailed compared to average velocity.

Time in Physics

Time is an essential scalar quantity in physics. It helps in measuring the interval or period during which events occur. Time has only magnitude and no direction, which means we often express time in units like seconds, minutes, or hours.
In our everyday and scientific endeavors, time remains a cornerstone for understanding motion. It helps determine how long an object has been in motion, which is vital for calculations involving displacement, speed, and velocity. For example, when calculating average speed or velocity, time serves as the denominator which scales the distance or displacement traveled.

• Time is always positive and progresses eternally forward, providing a constant reference for motion and change.
• The understanding of time is fundamental to the study of dynamics and kinematics.

Recognizing time as a baseline measurement in physics aids in analyzing various phenomena with clarity.