GRE Physics Companion: Tangential-Normal Particle Kinematics

This companion focuses on fast recognition of path-coordinate acceleration. The most useful relation is

(1)

with

(2)

1 Fast triage

If the problem asks how fast the particle’s speed is changing, use a_t.

If the problem gives a curved path and asks about the inward acceleration, use a_n.

If speed is given as a function of arc length,

(3)

2 Common traps

A particle can have a_t = 0 and still have nonzero acceleration if the path is curved.

The normal component always points toward the local center of curvature.

At fixed curvature radius,

(4)

At fixed speed,

(5)

3 Worked GRE example 1: constant-speed curve

A particle travels at 15 m/s around a path whose local radius of curvature is 45 m. Its speed is instantaneously constant.

Since

(6)

and

(7)

we have

(8)

directed toward the local center of curvature.

4 Worked GRE example 2: scaling

A particle moves along the same path, so ρ_c is unchanged. Its speed increases from v to 3v.

Because

(9)

the new normal acceleration is

(10)

Thus the normal acceleration increases by a factor of 9.

5 GRE-speed questions

1. A particle moves at constant speed along a curved path. Which component must be zero? (A) a_n (B) a_t (C) total acceleration (D) curvature.
2. At fixed radius of curvature, doubling speed changes a_n by a factor of (A) 1∕2 (B) 2 (C) 4 (D) 8.
3. At fixed speed, doubling ρ_c changes a_n by a factor of (A) 1∕2 (B) 1 (C) 2 (D) 4.
4. A particle has a_t = 3 m/s² and a _n = 4 m/s². Its total acceleration magnitude is (A) 1 (B) 5 (C) 7 (D) 12 m/s².

6 Answers and rationales

1. B. Constant speed means dv∕dt = 0, so a_t = 0; curvature can still produce a_n.
2. C. The normal component depends on v².
3. A. The normal component is inversely proportional to radius of curvature.
4. B. The components are perpendicular, so |a| = = 5 m/s².

References