Question

The ratio of pathlength and the respective time interval is (A) Mean Velocity (B) Mean speed (C) instantaneous velocity (D) instantaneous speed

Short answer

The given ratio \(\frac{\text{pathlength}}{\text{time interval}}\) represents (B) Mean speed, as it closely resembles the definition of mean speed, which involves the total distance (pathlength) divided by the time interval, without considering the direction of motion.

Step-by-step solution

Understand the given ratio

The question gives us a ratio: \(\frac{\text{pathlength}}{\text{time interval}}\). We need to find out which of the given concepts this represents.

Define mean velocity

Mean velocity is the total displacement divided by the total time interval. Displacement is the change in position and takes into account the direction of the movement.

Define mean speed

Mean speed is the total distance covered divided by the total time interval. Distance is the length of the path taken during the motion, and it doesn't consider the direction.

Define instantaneous velocity

Instantaneous velocity is the velocity of an object at a specific instant in time. It tells us how fast an object is moving while considering its direction.

Define instantaneous speed

Instantaneous speed is the magnitude of the instantaneous velocity, which means it only tells us how fast an object is moving but does not consider the direction.

Compare the given ratio to the definitions

In the given ratio, \(\frac{\text{pathlength}}{\text{time interval}}\), pathlength refers to the distance covered, and it doesn't consider the direction of the object. So, it can't be mean velocity or instantaneous velocity because both require considering the direction of motion (displacement). However, when we compare the ratio to the definitions of mean speed and instantaneous speed, we see that the given ratio closely resembles the definition of mean speed, as both involve the total distance (pathlength) divided by the time interval.

Determine the correct option

Based on our comparison, the ratio of pathlength and the respective time interval represents (B) Mean speed.

Key concepts

Velocity

Velocity is a fundamental concept in physics, representing how fast an object changes its position over time, including a specific direction. Imagine you're driving a car. If you travel north at 60 km/h, your velocity is expressed as 60 km/h north because it includes both speed and direction. The direction makes velocity a vector quantity, setting it apart from speed, which is scalar and only considers magnitude.
Velocity can be further categorized into different types:

• Average Velocity: This is computed by dividing the total displacement by the total time interval. Displacement, in this context, is the straight-line distance between starting and ending points, along with the direction.
• Instantaneous Velocity: This refers to the speed and direction of an object at a particular instant of time. We'll delve more into this later.

Remember, velocity requires both speed and direction. So, any change in either the speed or direction means a change in velocity.

Displacement

Displacement is the term used to describe the change in position of an object. It is a vector quantity, which means it includes both magnitude and direction. Consider displacement as the shortest path between two points, much like taking a straight line between your home and school, rather than following the winding roads you might normally travel. What makes displacement different from distance is its emphasis on the line used and the direction:[br]

• Magnitude: This aspect refers to the length of the straight path between two points.
• Direction: Unlike distance, displacement specifies the direction of motion, such as north, south, east, or west.

Displacement is crucial for calculating velocity, as it balances the notion of how quickly an object covers ground with the direction in which it’s moving.

Instantaneous Speed

Instantaneous speed is the speed of an object at a particular moment in time. It is similar to looking at your car's speedometer while driving - it tells you how fast you're going at that exact instant, not averaged over a longer period. While it doesn't include direction and is thus a scalar quantity, instantaneous speed provides valuable insight, especially in dynamic systems where the speed of an object can fluctuate rapidly. Here's why instantaneous speed is significant:

• It reflects real-time speed, which is critical for understanding how an object behaves at any given moment.
• In calculations involving instantaneous speed, we focus on just the magnitude, ignoring any directional considerations.

Instantaneous speed can vary considerably - for instance, if you're accelerating or decelerating, your speed can change drastically within a short span of time.

Distance

Distance refers to the total path length travelled by an object. Unlike displacement, distance does not care about direction. Consider it as the total ground covered during a trip, including any turns or curves taken. Suppose you take a walk around your neighborhood, plotting a course that loops around several blocks and back to where you started. The distance would be the entire length of the path you walked, even though your displacement could very well be zero if you returned to your starting point. A few important points about distance:

• Scalar Quantity: Since distance does not take direction into account, it's a scalar. It offers a complete measure of the "ground covered".
• Always Positive: As it measures total travelled length, it's non-negative by nature.
• Path Dependent: The actual path taken is crucial to the measurement of distance.

Understanding distance is key when working with concepts that involve the total length an object travels, as opposed to just the starting and ending point.