# Rates of change

Recall:

• $$s(t)$$ usually represents a position in one-dimension, either past zero (+) or behind zero (-)
• $$v(t) = s'(t)$$ usually represents velocity, moving forwards (+) or backwards (-)
• $$a(t) = v'(t) = s''(t)$$ usually represents acceleration, which can work with velocity (if signs match) or against (if signs are different)
• $$|v(t)|$$ represents the speed of an object and does NOT have a direction
• $$a'(t)$$ is called the "jerk", I do not know why, but it's funny so thought you should know

## 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 $$[0, 10]$$?