A graph is basically a picture of an equation. Graphs of an object's position, velocity, and acceleration as a function of time can tell you a great deal:

• The velocity is the slope of the position graph
• The displacement is the area under the curve of the velocity graph
• The acceleration is the slope of the velocity graph
• The change in velocity is the area under the curve of the acceleration graph

Predict what the position and velocity graphs look like for the red car, which has a constant velocity of 11.11 m/s.

Note that:

• Constant velocity means the velocity graph is horizontal, equal to 11.11 m/s at all times.
• A constant velocity means the position graph has a constant slope (of 11.11 m/s). It's a straight line sloping up, and starting below the origin.
• The displacement is the area under the curve of the velocity graph. Each second the car travels 11.11 m.

Predict what the acceleration graph looks like for the red car, which has a constant velocity of 11.11 m/s.

Note that:

• Constant velocity means the acceleration is zero.
• The change in velocity over a certain time interval equals the area under the acceleration graph over that interval. In this case the velocity does not change, so there can be no area under the acceleration graph.

Predict what the position and velocity graphs look like for the black car, which has a constant acceleration of 3 m/s² starting from rest.

Note that:

• Constant acceleration means the velocity graph has a constant slope.
• If the velocity steadily increases, the position graph must have a steadily increasing slope. Constant acceleration results in a parabolic position graph.
• Once again, the displacement is the area under the curve of the velocity graph.

Predict what the acceleration graph looks like for the black car, which has a constant acceleration of 3 m/s².

Note that:

• Constant acceleration means a horizontal line for the acceleration graph.
• The acceleration is the slope of the velocity graph. Constant acceleration = constant slope = straight line for the velocity graph.
• The area under the acceleration graph is the change in velocity.

Looking at the graphs for both cars at once can actually give us the answers to the sample problem we did earlier. This is another good way to check the answers.

The position graph shows that there are two locations where the cars pass each other. The red car passes the black one around 1.8 s, at x = 6 m, and the black one passes the red one around 5.7 s, at x = 50 m. This confirms our hard work earlier, solving the quadratic.

Finding the time when the black car passes the red one, you can then figure out how fast the black car is going at that time by reading it directly off the velocity graph. At t = 5.7 s, the black car's speed is about 17 m/s, about 50% more than the red car's speed at that time.