GRE Physics Companion: Position and Displacement

This entry is the GRE-oriented companion to M01-01. The core article develops position vectors, displacement, distance traveled, and coordinate dependence. Here the emphasis is rapid recognition and endpoint bookkeeping.

1 Fast test-day strategy

For ordinary position/ questions:

1. Identify the initial position and final position.
2. Compute displacement as final minus initial:

   (1)

3. If the question asks for distance traveled, add the lengths of the path segments instead.
4. For a closed trip, the total displacement is zero.
5. In two or three dimensions, subtract vector components, not vector magnitudes.
6. A constant shift of the coordinate origin changes position values but not the displacement between the same two points.

2 High-frequency traps

• Negative position does not imply negative distance.
• Distance traveled is never negative; one-dimensional displacement can be positive, negative, or zero.
• Returning to the starting point makes the net displacement zero, not the distance traveled.
• The magnitude |Δr| is the straight-line endpoint separation, not the length of a curved path.
• The magnitude of a position vector is distance from the chosen origin, not distance traveled during motion.

3 Worked GRE example 1: reversal on a line

A particle starts at x = −4 m, moves to x = 6 m, and then returns to x = 1 m. Find the total displacement and total distance traveled.

For displacement, use only the endpoints:

(2)

For distance traveled, add the lengths of both legs:

(3)

Thus

(4)

GRE shortcut: displacement ignores intermediate reversals; distance does not.

4 Worked GRE example 2: quarter-circle motion

A particle moves along a quarter circle of radius R from (R, 0) to (0,R). Which pair gives the distance traveled and displacement magnitude?

The arc length is

(5)

The endpoint displacement is the diagonal of a square with side R:

(6)

Therefore

(7)

5 GRE-speed questions

M01-01G-Q01

A particle moves from x = −4 m to x = 6 m. Its displacement is

(A) −10 m (B) −2 m (C) +2 m (D) +10 m

M01-01G-Q02

A student walks 3 m east and then 4 m north. The distance traveled and displacement magnitude are

(A) 5 m, 7 m (B) 7 m, 5 m (C) 7 m, 7 m (D) 5 m, 5 m

M01-01G-Q03

A runner completes one full circle of radius R and stops at the starting point. The total displacement magnitude is

(A) 0 (B) R (C) 2R (D) 2πR

M01-01G-Q04

A coordinate origin is shifted by a fixed vector while the axes remain parallel. Which quantity for motion between the same two physical points is unchanged?

(A) initial position vector (B) final position vector (C) displacement vector (D) distance from the new origin

M01-01G-Q05

A particle moves from (1, 2, 3) m to (4, 6, 3) m. The displacement magnitude is

(A) 3 m (B) 4 m (C) 5 m (D) 7 m

6 Answers and brief rationales

1. Q01: D. Δx = 6 − (−4) = 10 m.
2. Q02: B. Path length is 3+4 = 7 m; the straight endpoint separation is = 5 m.
3. Q03: A. Initial and final positions are identical.
4. Q04: C. A constant origin translation cancels when final minus initial is formed.
5. Q05: C. Δr = (3, 4, 0) m, whose magnitude is 5 m.

7 Test-day summary

The fastest reliable distinction is

(8)

while distance traveled follows the entire path. If a problem contains a reversal, curve, or closed loop, check immediately whether it is asking for a vector displacement or a scalar path length.

References