Which one of the following groups contains quantities that do not share the same dimensions?

Introduction / Context:
Dimensional analysis helps us understand whether two physical quantities are of the same kind. Many pairs like pressure and stress or work and energy are closely related and share dimensional formulas, while others differ. This question asks you to identify which group contains quantities that do not have the same dimensions, a common test of understanding in basic physics.

Given Data / Assumptions:

• Four pairs of quantities are given.
• We must find the pair that is dimensionally different.
• SI base dimensions mass (M), length (L), and time (T) are used.

Concept / Approach:
We need to write the dimensional formula for each member of the pair and compare. If they match, the pair has the same dimensions. If they do not match, that pair is the correct answer. Pressure and stress are both force per unit area. Velocity and speed are both displacement per unit time. Work and energy are both force times distance. However, force and impulse, although related, are not dimensionally identical because impulse includes an extra factor of time.

Step-by-Step Solution:
Step 1: Pressure and stress: both equal force per area, so [P] = [stress] = (M L T^-2) / L^2 = M L^-1 T^-2. Step 2: Velocity and speed: both are distance per time, so [v] = L T^-1 and speed has the same dimensions. Step 3: Work and energy: work W = force * distance, so [W] = M L T^-2 * L = M L^2 T^-2. Energy has the same dimensional formula. Step 4: Force and impulse: force F has [F] = M L T^-2, impulse J = F * time, so [J] = M L T^-2 * T = M L T^-1. These are different because of the exponent of T.

Verification / Alternative check:
Impulse is also defined as change in momentum, while force is rate of change of momentum. Since rate involves division by time, force will have one extra factor of T in the denominator compared to momentum or impulse. That confirms that force and impulse cannot share the same dimensions, even though they are closely related via Newton laws.

Why Other Options Are Wrong:
Pressure and stress: Identical definitions in mechanics mean they share the same dimensions. Velocity and speed: Speed is simply the magnitude of velocity; their dimensions are the same. Work and energy: In physics, work done on a system is stored as energy, so they naturally have the same dimensional form.

Common Pitfalls:
A common error is to focus only on how quantities are used in formulas and not on the definitions. Students may think force and impulse must have the same dimensions because both appear with similar symbols in mechanics. The key is to remember that impulse involves time multiplied by force, while force involves time in the denominator when defined from momentum. Always break down quantities to base units before deciding.

Final Answer:
The pair that does not have the same dimensions is force and impulse.