Which of the following have the same unit, ms-1 ?

Question

Prepp Team · Accepted Answer

Understanding Units in Physics

The question asks us to identify the pair of physical quantities that share the same unit, specifically given as \(\text{ms}^{-1}\). The unit \(\text{ms}^{-1}\) represents meters per second, which is a unit of speed or velocity. It describes how many meters an object travels in one second.

Analyzing the Units of Physical Quantities

Let's determine the standard SI units for each quantity mentioned in the options:

Speed: Speed is the rate of change of distance. Distance is measured in meters (m) and time in seconds (s). Therefore, the unit of speed is meters per second, or \(\text{m/s}\), which can also be written as \(\text{ms}^{-1}\).
Velocity: Velocity is the rate of change of displacement. Displacement is measured in meters (m) and time in seconds (s). Therefore, the unit of velocity is meters per second, or \(\text{m/s}\), which can also be written as \(\text{ms}^{-1}\).
Acceleration: Acceleration is the rate of change of velocity. Velocity is measured in \(\text{m/s}\) and time in seconds (s). Therefore, the unit of acceleration is (\(\text{m/s}\)) per second, or \(\text{m/s}^2\), which can also be written as \(\text{ms}^{-2}\).
Momentum: Momentum is the product of mass and velocity. Mass is measured in kilograms (kg) and velocity in \(\text{m/s}\). Therefore, the unit of momentum is kilogram meters per second, or \(\text{kg m/s}\), which can also be written as \(\text{kg ms}^{-1}\).

We can summarize the units in the following table:

Physical Quantity	SI Unit	Unit in \(\text{ms}^{-1}\) form
Speed	m/s	\(\text{ms}^{-1}\)
Velocity	m/s	\(\text{ms}^{-1}\)
Acceleration	\(\text{m/s}^2\)	\(\text{ms}^{-2}\)
Momentum	kg m/s	\(\text{kg ms}^{-1}\)

Evaluating the Given Options

Now, let's examine each option based on the units we determined:

Option 1: Speed and momentum
- Unit of Speed: \(\text{ms}^{-1}\)
- Unit of Momentum: \(\text{kg ms}^{-1}\)
- The units are different.
Option 2: Acceleration and momentum
- Unit of Acceleration: \(\text{ms}^{-2}\)
- Unit of Momentum: \(\text{kg ms}^{-1}\)
- The units are different.
Option 3: Velocity and acceleration
- Unit of Velocity: \(\text{ms}^{-1}\)
- Unit of Acceleration: \(\text{ms}^{-2}\)
- The units are different.
Option 4: Speed and velocity
- Unit of Speed: \(\text{ms}^{-1}\)
- Unit of Velocity: \(\text{ms}^{-1}\)
- The units are the same.

Based on this analysis, only the pair "Speed and velocity" have the same unit, which is \(\text{ms}^{-1}\).

Conclusion

The pair of physical quantities that have the same unit, \(\text{ms}^{-1}\), is Speed and velocity. Both represent a rate of motion, although velocity is a vector quantity (includes direction) while speed is a scalar quantity (magnitude only).

Revision Table: Units of Common Quantities

Quantity	Definition	Formula	Unit (\(\text{ms}^{-1}\) form)	Type (Scalar/Vector)
Speed	Rate of distance covered	\( \text{Distance / Time} \)	\(\text{ms}^{-1}\)	Scalar
Velocity	Rate of displacement	\( \text{Displacement / Time} \)	\(\text{ms}^{-1}\)	Vector
Acceleration	Rate of change of velocity	\( \text{Change in Velocity / Time} \)	\(\text{ms}^{-2}\)	Vector
Momentum	Product of mass and velocity	\( \text{Mass} \times \text{Velocity} \)	\(\text{kg ms}^{-1}\)	Vector

Additional Information: Scalar vs. Vector Quantities

Understanding whether a quantity is scalar or vector is important in physics, even if they share the same unit.

Scalar Quantity: A quantity that is completely described by its magnitude alone. Examples include speed, mass, distance, time, and temperature.
Vector Quantity: A quantity that is completely described by both its magnitude and direction. Examples include velocity, displacement, acceleration, force, and momentum.

Speed and velocity both measure how fast an object is moving (magnitude), but velocity also tells us which direction it's moving in. Despite this difference in nature (scalar vs. vector), their fundamental units based on length and time are the same (\(\text{ms}^{-1}\)).