Rates of change

Recall:

Speeding up/down

An object starts by moving at a given velocity and acceleration. Complete the velocities below after the indicated time intervals.

v(t) a(t) 1 sec 2 secs 3 secs 4 secs 5 secs Speed up/down?
5 m/s -1 m/s/s
0 m/s/s
1 m/s/s
-5 m/s -1 m/s/s
0 m/s/s
1 m/s/s

Conjectures

A. An object is speeding up when:


B. An object is slowing down when:


Practice

1. Let \( s(t) = t^4 - 8t^3 + 18t^2 \) be the straight-line position of an object at \( t \) (time).

  1. Find the velocity at \( t=2 \) and \( t=4 \).
  2. Find the acceleration at \( t=2 \) and \( t=4 \).
  3. Determine when the object is at rest.
  4. Determine when the object has no acceleration.
  5. Over what time intervals is the object speeding up? Slowing down?

2. Let \( s(t) = e^t (t-5) \) be the straight-line position of an object at \( t \) (time).

  1. Find the velocity at \( t=0 \) and \( t=5 \).
  2. When is the object at rest?
  3. When is the object moving backwards?
  4. When is the object speeding up?

Challenge: Distance vs Displacement

Example: If I go forward 10 miles, backwards 3 miles, then forward 5 miles, my distance traveled is 18 miles. However, my displacement is 12 miles from where I started. Distance is always positive.

Let \(s(t) = (t-5)(t-10) = t^2 - 15t + 50 \) be the straight-line position of an object at time \(t\).

  1. When does the object change directions?
  2. Find the total distance traveled on the interval \( [0, 10] \). Hint: How far did the object move backward? How far did the object move forward?
  3. What is the displacement of the object over the interval \( [0, 10] \)?