Questions: Curvilinear Motion: Tangential and Normal Components

Question 1

A car travels around a circular track at a perfectly constant speed. Which statement correctly describes its acceleration?

• A. Acceleration is zero because speed is not changing
• B. Acceleration is entirely tangential because the car is maintaining a constant rate of motion
• C. Acceleration is entirely normal (centripetal), pointing toward the center of the circle, because speed is constant but direction is continuously changing
• D. Acceleration is zero in both components because circular motion at constant speed requires no net force

Answer: (C)

Acceleration is the rate of change of the velocity vector, not just speed. Even at constant speed, the velocity direction continuously rotates as the car follows the curve — and a rotating vector has a nonzero derivative. Since speed is constant (a_t = dv/dt = 0), all acceleration comes from the changing direction, which is the normal component a_n = v²/ρ directed toward the center. Newton's second law then requires a centripetal force — no force, no curve.

Question 2

A particle moves along a curved path where the radius of curvature ρ = 50 m and its speed doubles from 10 m/s to 20 m/s. By what factor does the normal acceleration change?

• A. It doubles, because speed doubled
• B. It quadruples, because a_n = v²/ρ and speed appears squared
• C. It remains the same, because ρ did not change
• D. It halves, because the tangential acceleration increased

Answer: (B)

The normal acceleration formula is a_n = v²/ρ. Doubling speed while holding ρ constant gives a_n' = (2v)²/ρ = 4v²/ρ = 4·a_n. The quadratic dependence on speed is crucial: a small increase in speed has a disproportionately large effect on centripetal acceleration (and thus on the centripetal force required). This is why highway curves must be designed with generous radii — at higher speeds, the required centripetal force grows with the square of speed.

Question 3

An object moving along a curve with decreasing speed has zero tangential acceleration.

• T. True
• F. False

Answer: False

Tangential acceleration a_t = dv/dt is the rate of change of speed. If speed is decreasing, then dv/dt is negative — the tangential acceleration is nonzero and directed opposite to the direction of motion. Zero tangential acceleration means constant speed, not zero motion. A car braking on a curve has negative (decelerating) tangential acceleration and nonzero normal acceleration from the curve simultaneously.

Question 4

The normal acceleration component always points toward the center of curvature, regardless of the direction of motion along the path.

• T. True
• F. False

Answer: True

The normal direction e_n is defined as pointing toward the center of curvature — the center of the osculating circle that best fits the path at that point. This is always perpendicular to the velocity and always directed inward, regardless of whether the particle is speeding up, slowing down, or traveling in either direction along the path. The geometry of the path determines e_n, not the dynamics of the particle.

Question 5

Why does an object moving at constant speed around a curve still experience acceleration, and what physical force must be providing it?

Model answer: Acceleration is defined as the rate of change of the velocity vector, not just its magnitude. Even at constant speed, the velocity direction continuously changes as the object follows the curve — this constitutes a nonzero rate of change of the velocity vector. The normal component a_n = v²/ρ captures this purely directional change, always pointing toward the center of curvature. By Newton's second law, a nonzero acceleration requires a nonzero net force in that direction. The physical force providing this inward acceleration is whatever causes the curvature — friction between tires and road, tension in a string, gravity for orbital motion, the normal force from a banked track, etc.

The deep insight is that 'constant speed' is not the same as 'not accelerating.' Students who conflate speed with velocity miss this, leading to the incorrect conclusion that circular motion at constant speed requires no force. The normal acceleration component exists whenever the path curves, regardless of whether speed is changing.