## We all understand speed intuitively. Velocity and acceleration are more mathy and harder for many students to comprehend, especially on a graph. Even harder is the idea of when an object is *slowing down* or *speeding up*. In this post I attempt to unpack both.

Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling (think of backing up). Both cars below are traveling at equivalent speeds.

Velocity can be graphed (the red curve below). Here is information found in the graph.

- Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left.
- The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed).
- The blue lines are tangent lines with slope equal to acceleration (the derivative of velocity). If the line is going down from left to right, the acceleration is negative. Up, positive.

Below is an example of using prior knowledge and visuals to help make sense of speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. When the car is above the x-axis, it has positive velocity. Below, negative. When the car is headed down, the acceleration is negative. Up, positive.

Now for speeding up or slowing down.

- Think of a bank account balance.
- If you have a positive balance and a transaction that is negative, Your balance is moving towards 0.
- Suppose you have a negative balance in your bank account and you have a positive transaction. Your balance is moving towards 0.

- As the car moves towards a stop sign it will slow down. The speed is getting closer to 0.
- Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0, like the bank balance analogy.
- Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0.

- As the car moves away from a stop sign it speeds up. The speed is moving away from 0.
- Positive bank balance, positive transaction means balance is moving away from 0.
- Negative bank balance, negative transaction means balance is moving away from 0.