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write-dimensions-of-velocity

๐Ÿš€ Dimensional analysis is a technique used in physics and engineering to convert units from one system to another and to check the consistency of equations. Velocity is defined as the rate of change of displacement with respect to time. In dimensional analysis, we express the dimensions of physical quantities in terms of fundamental dimensions: mass (M), length (L), and time (T). The dimension of velocity is derived from its definition: it is displacement (length) divided by time. Therefore, the dimensions of velocity can be expressed as [V] = [L]/[T]. This means that velocity has dimensions of length per unit time, such as meters per second (m/s). Understanding the dimensions of velocity is crucial for solving problems in kinematics and dynamics, and for ensuring that equations involving velocity are dimensionally consistent.

Theory Explanation

Step 1: Understanding Displacement and Time

Displacement is a vector quantity that refers to the change in position of an object. It is measured in units of length, such as meters (m). Time is a scalar quantity that measures the duration of an event, measured in seconds (s). To find the dimensions of velocity, we need to relate these two quantities.

Step 2: Definition of Velocity

Velocity is defined as the rate of change of displacement with respect to time. Mathematically, it is expressed as: V = ฮ”x/ฮ”t, where ฮ”x is the change in displacement and ฮ”t is the change in time. This definition is fundamental to deriving the dimensions of velocity.

\[ V = \frac{\Delta x}{\Delta t} \]
Step 3: Deriving Dimensions of Velocity

To find the dimensions of velocity, we replace displacement (ฮ”x) with its dimension [L] (length) and time (ฮ”t) with its dimension [T]. Therefore, the dimensions of velocity can be expressed as: [V] = [L]/[T]. This gives us the dimensional formula for velocity.

\[ [V] = \frac{[L]}{[T]} \]

Key Points

  • ๐ŸŽฏ Velocity is defined as displacement per unit time.
  • ๐ŸŽฏ The dimensional formula for velocity is [L]/[T].
  • ๐ŸŽฏ Understanding dimensions helps in verifying the consistency of physical equations.

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Examples:💡

Example 1: Determine the dimensions of velocity given the formula V = s/t, where s is distance and t is time.

Solution:

Step 1: Identify the dimensions of distance (s), which is length. Therefore, [s] = [L].

\[ [s] = [L] \]

Step 2: Identify the dimensions of time (t), which is measured in seconds. Therefore, [t] = [T].

\[ [t] = [T] \]

Step 3: Substitute the dimensions into the formula for velocity: V = s/t. This gives us [V] = [L]/[T].

\[ [V] = \frac{[L]}{[T]} \]

Example 2: If a car travels 100 meters in 5 seconds, find the dimensions of its velocity.

Solution:

Step 1: Identify the distance traveled, which is 100 meters. Therefore, [s] = [L].

\[ [s] = [L] \]

Step 2: Identify the time taken, which is 5 seconds. Therefore, [t] = [T].

\[ [t] = [T] \]

Step 3: Using the formula for velocity, V = s/t, we find [V] = [L]/[T].

\[ [V] = \frac{[L]}{[T]} \]

Common Mistakes

  • Mistake: Students often confuse velocity with speed, forgetting that velocity is a vector quantity and speed is a scalar.

    Correction: Remember that velocity has both magnitude and direction, while speed only has magnitude.

  • Mistake: Some students forget to convert all units to the same system before performing calculations.

    Correction: Always ensure that all measurements are in compatible units before applying the dimensional analysis.

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