Which statement correctly describes the relationship between displacement and total distance traveled over [a,b]?

Which statement correctly describes the relationship between displacement and total distance traveled over [a,b]?

• A. Displacement over [a,b] is ∫_a^b v(t) dt; Total distance is ∫_a^b |v(t)| dt
• B. Displacement is ∫_a^b |v(t)| dt; Total distance is ∫_a^b v(t) dt
• C. Displacement equals ∫_a^b a(t) dt; Total distance equals ∫_a^b v(t) dt
• D. Displacement equals ∫_a^b v(t) dt; Total distance equals ∫_a^b a(t) dt

Answer: (A)

The important idea is that displacement tracks only the net change in position, while total distance adds up all movement regardless of direction. The displacement over an interval is found by integrating the velocity: ∫_a^b v(t) dt. The total distance traveled is the integral of speed, which is the magnitude of velocity: ∫_a^b |v(t)| dt. Because speed is always nonnegative, this integral Never cancels out like displacement can when velocity changes sign. This is why the statement pairing displacement with ∫ v(t) dt and total distance with ∫ |v(t)| dt is the correct one. If velocity never changes sign, displacement and total distance are the same in magnitude, but when velocity switches direction, displacement can be smaller than the total distance because the backward and forward motions partly cancel in displacement but add in distance.

The important idea is that displacement tracks only the net change in position, while total distance adds up all movement regardless of direction. The displacement over an interval is found by integrating the velocity: ∫_a^b v(t) dt. The total distance traveled is the integral of speed, which is the magnitude of velocity: ∫_a^b |v(t)| dt. Because speed is always nonnegative, this integral Never cancels out like displacement can when velocity changes sign.

This is why the statement pairing displacement with ∫ v(t) dt and total distance with ∫ |v(t)| dt is the correct one. If velocity never changes sign, displacement and total distance are the same in magnitude, but when velocity switches direction, displacement can be smaller than the total distance because the backward and forward motions partly cancel in displacement but add in distance.