Q.Pick out the only vector quantity in the following list: Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

Concept: Scalar vs Vector quantities

A scalar has only magnitude (a number with units). A vector requires both magnitude and direction to be fully specified.

Examining each quantity:

• Temperature, pressure, time, power, energy, gravitational potential, charge: pure magnitudes with no inherent direction.
• Total path length: the distance traveled, a scalar (contrast with displacement, which is a vector).
• Coefficient of friction: a dimensionless ratio, purely scalar.
• Impulse: defined as $\vec{J} = \int \vec{F} \, dt = \Delta \vec{p}$, the change in momentum. Since force and momentum are vectors, impulse inherits direction—it points along the direction of the net force applied.

Understanding Scalar vs Vector Quantities

Physical quantities divide into two fundamental categories based on how completely they describe a phenomenon.

A scalar is fully specified by a single number (with units). Temperature of 300 K, energy of 50 J, or charge of $-1.6 \times 10^{-19}$ C tells you everything about that quantity. Scalars add algebraically: if you have 3 J of energy here and 5 J there, the total is simply 8 J.

A vector requires both magnitude and direction for complete description. A force of 10 N means nothing until you specify which way it acts. Vectors follow special addition rules (the parallelogram law) because direction matters: two 10 N forces can produce anything from 0 N (opposite directions) to 20 N (same direction).

The test is simple: Does the quantity have an inherent direction in space?

Examining Each Quantity

Let me walk through the list systematically:

1. Temperature – A measure of average kinetic energy of molecules. 30°C is 30°C everywhere in a uniform body; there's no "direction" to temperature. Scalar.

2. Pressure – Force per unit area, but it acts equally in all directions at a point in a fluid (Pascal's principle). Though force is a vector, pressure itself has no preferred direction. Scalar.

3. Impulse – Defined as $\vec{J} = \int \vec{F} \, dt = \Delta \vec{p}$. Since force $\vec{F}$ is a vector and time is scalar, impulse inherits the direction of the net force. Equivalently, impulse equals the change in momentum, which is definitely a vector. Vector.

4. Time – Duration between events. 5 seconds is 5 seconds; it doesn't point anywhere. Scalar.

5. Power – Rate of energy transfer, $P = \frac{dE}{dt}$. Energy is scalar, time is scalar, so their ratio is scalar. Scalar.

6. Total path length – The actual distance traveled along a path. Even if you walk in circles, path length just accumulates as a number. (Contrast with displacement, which is a vector.) Scalar.

7. Energy – Capacity to do work. 100 J of kinetic energy or potential energy is just a magnitude. Scalar.

8. Gravitational potential – Energy per unit mass at a point in a field, $V = -\frac{GM}{r}$. A single number at each location. Scalar.

9. Coefficient of friction – A dimensionless ratio $\mu = \frac{f}{N}$ relating magnitudes of friction to normal force. Pure number, no direction. Scalar.

10. Charge – Amount of electric charge. $+2 \times 10^{-6}$ C or $-5$ C is complete information. (The electric field created by charge is a vector, but charge itself is not.) Scalar.

Concept: Grouping the Scalars by Why They're Scalars, Then Isolating the One Vector

Method: Categorize First, Then Single Out the Odd One Out

Rather than examining all ten quantities in one flat list, this method groups the nine scalars into a few small categories that share the same reason for being scalar, so the genuine vector (impulse) stands out clearly as not fitting any category.

Category 1 -- Pure magnitudes / state variables (no formula needed, no direction is even conceivable)

Temperature, time, charge. Each is a single number describing "how much" of something; asking "which direction is 300 K pointing?" is meaningless.

Category 2 -- Ratios or products of two scalars (a scalar built from scalars stays a scalar)

Power ($P = dE/dt$, scalar/scalar), coefficient of friction ($\mu = f/N$, scalar/scalar -- even though $f$ and $N$ are magnitudes of vector forces, their ratio is dimensionless and directionless), gravitational potential ($V=-GM/r$, built from scalars $G,M,r$).

Category 3 -- Quantities that act isotropically or are path-integrals of scalars

Pressure (acts equally in all directions at a point -- Pascal's principle -- so it has no single preferred direction of its own), energy (a capacity to do work, expressed as one number), total path length (the accumulated distance travelled, which keeps adding regardless of which way the object turns).

The one quantity that doesn't fit any category -- Impulse

Trace impulse to its defining formula:

Force $\vec F$ is a vector, and integrating a vector over a scalar variable ($t$) does not remove its direction -- the result $\vec J$ is a vector. Equivalently, impulse equals the change in momentum $\Delta\vec p$, and momentum is unambiguously a vector. Unlike every quantity in Categories 1-3, impulse's defining formula has a vector ($\vec F$ or $\vec p$) surviving undivided and un-dotted into the final result.

Why grouping first makes the answer robust

If each of the nine scalars is checked individually, it's easy to second-guess one (e.g. "does pressure have a direction, since it comes from force?"). Grouping by why a quantity is scalar -- pure magnitude, scalar ratio, or isotropic/path-accumulated -- shows that none of the nine reasons ever lets a vector survive into the final quantity, which is exactly the property that impulse's own formula violates.

Final Answer

The only vector quantity in the list is $\boxed{\text{impulse}}$.