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Distance Traveled Problems for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

Summary of Formulas

Position Function: s (t); s (t) = v(t) dt.

Velocity: v(t) = ; v(t) = v(t) dt.

Acceleration: a(t) = .

Speed: |v(t)|.

Displacement from t)1 to t)2 = v(t) dt = s (t2)–s (t1).

Total Distance Traveled from t1 to t2 = |v(t)| dt.

Example 1

Distance Traveled Problems

The graph of the velocity function of a moving particle is shown in Figure 13.2-1. What is the total distance traveled by the particle during 0 ≤ t ≤ 12?

Total Distance Traveled

Example 2

The velocity function of a moving particle on a coordinate line is v(t) = t2 +3t – 10 for 0 ≤ t ≤ 6. Find (a) the displacement by the particle during 0 ≤ t ≤ 6, and (b) the total distance traveled during 0 ≤ t ≤ 6.

The total distance traveled by the particle is or approximately 88.667.

Example 3

The velocity function of a moving particle on a coordinate line is v(t) = t3 – 6t2 +11t – 6. Using a calculator, find (a) the displacement by the particle during 1 ≤ t ≤ 4, and (b) the total distance traveled by the particle during 1 ≤ t ≤ 4.

Example 4

The acceleration function of a moving particle on a coordinate line is a(t) = – 4 and v0 =12 for 0 ≤ t ≤ 8. Find the total distance traveled by the particle during 0 ≤ t ≤ 8.

Total distance traveled by the particle is 68.

Example 5

The velocity function of a moving particle on a coordinate line is v(t)=3 cos(2t) for 0 ≤ t ≤ 2π. Using a calculator:

  1. Determine when the particle is moving to the right.
  2. Determine when the particle stops.
  3. The total distance traveled by the particle during 0 ≤ t ≤ 2π.

Solution:

  1. The particle is moving to the right when v(t) > 0. Enter y 1= 3 cos(2x ). Obtain y1 =0 when t = The particle is moving to the right when:
  2. The particle stops when v(t) = 0.
  3. Thus the particle stops at t =

  4. Total distance traveled |3 cos(2t)| dt.
  5. Enter (abs(3 cos(2x )), x, 0, 2π) and obtain 12. The total distance traveled by the particle is 12.

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

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