Dimensional Analysis

Step 1: Let the common distance of the race be . Since both cars start from rest with constant acceleration: where and are the times taken by cars A and B respectively.
Step 2: Given that car A finishes seconds earlier:
Step 3: From the distance equation:
Step 4: Write and substitute:
Step 5: Solving gives:
Step 6: Final speeds at the finish line:
Step 7: Given : Substituting and simplifying gives:

Step 1: Recall a fundamental principle of dimensional analysis: the exponent of an exponential function must be dimensionless. Since the exponent is given as , its dimensions must satisfy
Step 2: Writing this in terms of dimensions, we have Hence,
Step 3: Therefore, the dimension of the constant is the inverse of time squared. Hence → (B).

Step 1: Recall the definition of friction. Friction is a resistive force that opposes the relative motion or the tendency of motion between two surfaces in contact. Step 2: Determine the direction of frictional force. When an object is moving or trying to move on a surface:

Friction always acts opposite to the direction of motion (or impending motion).
Step 3: Analyze the options.

(A) Incorrect — friction does not necessarily act in the direction of applied force
(B) Incorrect — friction never acts in the same direction as motion
(C) Correct — friction acts opposite to motion ✔
(D) Incorrect — a correct option exists
Step 4: Final conclusion. The correct answer is .