For a body moving in a straight line with uniform (constant) acceleration, its final velocity can be expressed in terms of its average velocity and initial velocity by which of the following relations?

Introduction / Context:
In uniformly accelerated motion along a straight line, there are several useful kinematic equations that relate displacement, time, initial velocity, final velocity, and acceleration. One of these relations connects the average velocity to the initial and final velocities. From that, we can also express the final velocity in terms of the average and initial velocities. This question tests your understanding of these basic kinematic relationships.

Given Data / Assumptions:

• Motion is along a straight line.
• Acceleration is uniform (constant) during the time interval considered.
• We have an initial velocity u, a final velocity v, and an average velocity v_avg.
• We want an expression for v in terms of v_avg and u.

Concept / Approach:
For uniformly accelerated motion in a straight line, the average velocity over a time interval is simply the arithmetic mean of the initial and final velocities: v_avg = (u + v) / 2 From this equation, we can solve for v in terms of v_avg and u by rearranging. This algebraic manipulation gives a direct formula for final velocity in the desired form, which we can then compare with the options.

Step-by-Step Solution:
Step 1: Start with the relationship v_avg = (u + v) / 2. Step 2: Multiply both sides by 2 to remove the denominator: 2 * v_avg = u + v. Step 3: Rearrange to solve for v: v = 2 * v_avg - u. Step 4: Express this in words: final velocity equals two times the average velocity minus the initial velocity. Step 5: Compare this expression with the options and choose the one that matches.

Verification / Alternative check:
As a quick check, imagine a case where the initial velocity u = 0 and the final velocity v is some positive value. In uniformly accelerated motion, the average velocity would then be v_avg = (0 + v) / 2 = v / 2. Applying our relation v = 2 * v_avg - u gives v = 2 * (v / 2) - 0 = v, which is correct. This confirms that the algebra and the relationship are consistent with a simple example.

Why Other Options Are Wrong:

• Final velocity = average velocity minus initial velocity: This does not follow from the equation v_avg = (u + v) / 2 and gives wrong values when tested with numerical examples.
• Final velocity = 2 × average velocity plus initial velocity: This would give v = 2 * v_avg + u, which is inconsistent with the derivation and leads to an incorrect relationship.
• Final velocity = average velocity plus initial velocity: This relation has no basis in standard kinematic equations and is dimensionally misleading in this context.

Common Pitfalls:
A common error is to misremember the formula for average velocity, sometimes writing v_avg = (v - u) / 2 instead of (u + v) / 2. Another frequent mistake is incorrect algebra when rearranging equations, such as forgetting to multiply both sides by 2 or incorrectly isolating v. Careful step-by-step manipulation and checking with simple test values help avoid these issues.

Final Answer:
For uniformly accelerated motion, the final velocity is final velocity = 2 × average velocity minus initial velocity.