6.1 Velocity and Net Change

Displacement and Distance Traveled

In calculus, integration is used to analyze motion along a straight line by relating position, velocity, and acceleration. The displacement of an object is the net change in its position over a time interval, while the distance traveled is the total length of the path, regardless of direction.

• Displacement: If s(t) is the position function, the displacement over [a, b] is s(b) - s(a).

• Distance Traveled: The total distance is the integral of the speed (the absolute value of velocity):

• Velocity and Position: Velocity is the derivative of position: v(t) = s'(t).

• Fundamental Theorem of Calculus (FTC): Relates the integral of velocity to displacement:

Example: Jogger's Motion

Consider a jogger with velocity function v(t) = 2t^2 - 8t + 6 for 0 \leq t \leq 3 (in mi/hr). We analyze the motion as follows:

• Direction of Motion: The jogger moves in the positive direction when v(t) > 0 and in the negative direction when v(t) < 0.

• Displacement on Intervals:

  • On [0, 1]:

  • On [1, 3]:

  • On [0, 3]:

• Distance Traveled:

Position from Velocity

Given the velocity function and initial position, the position at any time t is found by integrating velocity:

• Theorem 6.1:

Example: If v(t) = \frac{1}{4} \sin t, s(0) = -\frac{1}{4}, then

Piecewise Velocity Example

For a skydiver with a piecewise velocity function, the total distance is the sum of integrals over each interval:

Velocity from Acceleration

Given acceleration a(t) and initial velocity v(0), the velocity is:

Example: Projectile Motion

An artillery shell is fired upward with v(0) = 300 m/s from s(0) = 30 m. Gravity acts with a(t) = -9.8 m/s2.

• Velocity:

• Position:

• Highest Point: Occurs when s, m

• Time to Ground: Solve s

6.1 Net Change and Future Value

Net Change Theorem

The Net Change Theorem states that the net change in a quantity Q(t) over [a, b] is the integral of its rate of change:

• Future value from initial value:

Example: Population Growth

A town has initial population 250 and grows at rate .

• Net Change (10 to 20 years):

• Population at time t:

Summary Tables