1. Velocity Time Graphs

2. What do Position Time graphs show? They show how position changes with time. So far we have studied graphs that show uniform motion. Or a constant velocity. We have seen some graphs that change slope, but they do this instantaneously. This is an unrealistic way to view the world.

3. What does a changing d t graph look like? This we don’t have the tools to work with a graph like this it varies too much.

4. What kinds of changing d t graphs can we handle? Consistent curves.

5. How do we read a curve on a d t graph? Look at how the slope is changing. The best way to do this is to find the slope at several points along the curve. The slope of a curve at any point can be found by drawing a tangent at that point.

6. The slope of a Tangent Tangent is any line that just touches the curve at that point. Looking at the tangent line you can tell that the slope is increasing from point 1 to point 3 1 2 3

7. What does a changing slope on a d t graph mean? Slope on a d t graph is the velocity of the object. So a changing slope means a changing velocity! This graph has an increasing slope so that means the velocity is increasing or has a positive change. This graph has a decreasing slope so that means the velocity is decreasing or has a negative change.

8. What about these? These can blow your mind unless you apply the tangent model. This graph starts off with a steep negative slope and goes to a shallower slope. The velocity has a high negative value and goes to a smaller negative value. This is a positive change in velocity. This graph starts off with a shallow negative slope and goes to a steep negative slope. The velocity has a low negative value and goes to a high negative value. This is a negative change in velocity.

9. Now what? These curved graphs are fine to understand the motion qualitatively however, they are not very useful quantitatively. Since these graphs are continuously or consistently changing. Then they too, are a form of uniform motion. A uniformly changing motion. Lets see how a d t graph translates into a v t graph.

10. Exploring a v t Graph What does a linear d t graph look like as a v t? v → (m/s) t (s) 4 5 d → (m) t (s) 20 5 Slope = 4 m/s A constant slope means a constant or unchanging velocity. That translates into a horizontal v t graph.

11. A changing d t graph If the d t graph is uniformly changing in the positive direction. You can tell the change by looking at how the tangent changes! Then the velocity is consistently changing. Which translates into a positively sloped straight line. Positively changing velocity Positive acceleration D (m) V (m/s) Horizontal = 0 velocity Steep slope = high velocity Starting at 0 velocity Ending at a high velocity

12. A changing d t graph If the d t graph is uniformly changing in the positive direction. Then the velocity is consistently changing. Which translates into a positively sloped straight line. When velocity reaches maximum value it remains constant Constant velocity means 0.00 acceleration = horizontal line Positively changing velocity Positive acceleration D (m) V (m/s) Constant velocity

13. A changing d t graph If the d t graph is uniformly changing in the negative direction. You can tell the change by looking at how the Tangent changes! Then the velocity is consistently changing negatively. Which translates into a negatively sloped straight line. Negatively changing velocity Negative acceleration D (m) V (m/s) Vertical = High velocity Low slope = low velocity Starting at a high velocity Ending at a low velocity

14. A negatively changing d t graph If the d t graph is uniformly changing in the negative direction. You can tell how it is changing by looking at the Tangent. Then the velocity is consistently changing. Which translates into a negatively sloped straight line. XX

15. Yes, we have a straight line again! We have a burning desire to calculate the slope. Slope of a v t graph would be…. Slope = rise = ∆v run ∆t This value will measure the rate of change of velocity with time. This is known as acceleration!

16. Exploring Acceleration This term describes how velocity changes with time. Acceleration is a vector quantity. Units:mm x 1m s = s s= s 2 s What does 5.0 m/s 2 mean? It means that the velocity changes by + 5.0 m/s every second.

17. What about deceleration? Avoid using this term altogether! Why? Deceleration indicates slowing down toward zero velocity. This would also inherently imply a negative velocity. This is not always true!

18. Lets See This is deceleration on a v t graph. This object has a decreasing velocity ending at zero.

19. What happens here? If we extend the deceleration graph beyond the zero velocity the object has an increasing velocity in the negative direction. This is NOT what deceleration means. Deceleration Acceleration in the negative direction.

20. What happens here? If there is a negative velocity to decelerate the object needs to have a positive change in velocity to reach zero velocity. This is a positive deceleration which doesn’t make sense! Deceleration (positive) Acceleration

21. Solution Always use acceleration: the rate of change of velocity with time. A vector quantity. Acceleration in the negative direction. - a Acceleration in the positive direction. + a

22. Assignment Complete V T handout assignment