Kinematics

1. Kinematics Prepared by: C. Richards

2. What is Kinematics? • Kinematics is the study of motion

3. The Difference Between Distance and Displacement • Distance is the total distance an object moves during a particular journey. It is usually measured in metres or kilometres • Displacement is the distance from the starting point at a particular instant in time. It is usually measured in metres or kilometres

4. Displacement vs. Distance • Distance and displacement are different. When you travelled 50 km to the East and then 20 km to the West, the total distance you travelled is 70 km, but your displacement is 30 km East.

5. Speed • Speed is a measure of how fast an object is moving. It tells how much distance you cover in a particular time. Speed = distance travelled time elapsed The unit for speed is determined by the unit of distance and time used in the above formula. So if distance is in meters and time is in seconds, then speed is measured in m/s or ms-1. Similarly, if distance is measured in km and speed in hr, then speed is measured in km/hr or kmh-1.

6. Velocity • Velocity is also a measure of how fast an object is moving, BUT it tells how fast it moves in ONE direction. Velocity = displacement time elapsed Velocity has the same unit of measurement as speed, i.e. ms-1 or kmhr-1

7. Putting speed and velocity into perspective • I am driving at a speed of 30km/hr. I leave Spanish Town at 8am to get to Kingston which is 75 km away. At what time will I get to Kingston? Solution: Every 1 hour I will complete 30 km After x hours I will complete 75 km Therefore, x = 75 ÷ 30 = 2.5 = 2hrs 30mins So, I will arrive in Kingston at 10:30am

8. Distance-time graphs Speed! The gradient of a distance-time graph gives us This general graph represents the motion of a body travelling at constant speed. The graph is linear (that is, a straight line).

9. Displacement – time graph • The gradient of the displacement-time graph gives us Velocity! If we have a high velocity, the graph has a steep slope. If we have a low velocity the graph has a shallow slope (assuming the vertical and horizontal scale of each graph is the same).

10. Example • A marathon runner runs at a constant 12 km/h. • a. Express her displacement travelled as a function of time. • b. Graph the motion for 0 ≤ t ≤ 4 h Solution: (a) Displacement = 12t , where t represents time

11. The graph is a straight line because her velocity was constant. That means, the gradient was constant for the 4 hours of running. So that means the runner never stopped, slowed down or sped up for the 4 hours What would the graph look like if the body in motion was not as consistent?

12. Example 2 Below is the graph of a journey by a car Find the velocity for each part of the journey.

13. Solution • For t = 0 to 2: • For t = 2 to 7: • For t = 7 to 8: The velocity is negative, since the particle is going in the opposite direction

14. Acceleration • Acceleration is the change in velocity per time. Acceleration = change in velocity change in time • A common unit for acceleration is ms-2. An acceleration of 7 ms-2 means that in each second, the velocity increases by 7 ms-1 (also written as 7 m/s).

15. Velocity – time graph Gradient = change in y (velocity) change in x (time) = acceleration • The area under the graph on the velocity-time graph gives displacement, while the area under the speed-time graph gives distance

16. Consider the graphs below and tell if the object has positive, negative or zero acceleration and velocity Positive VelocityZero Acceleration Positive VelocityPositive Acceleration

17. Velocity-time graphs (cont’d) • If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration is negative, then the slope is negative (i.e., a downward sloping line).

18. An object is moving in the positive direction if the line is located in the positive region of the graph (whether it is sloping up or sloping down). And one knows that an object is moving in the negative direction if the line is located in the negative region of the graph (whether it is sloping up or sloping down). And finally, if a line crosses over the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.

20. Speeding up or slowing down Speeding up means that the magnitude (or numerical value) of the velocity is getting larger. So, if the line is getting further away from the x-axis (the 0-velocity point), then the object is speeding up. And conversely, if the line is approaching the x-axis, then the object is slowing down.

21. Consider the graph below: Describe the motion of the object that exhibits the graph above.

22. Based on the graph, going from left to right, the gradient is positive and gets less steep until it becomes zero (at max of graph) then it becomes negative. So the velocity is increasing at a decreasing rate up, then starts decreasing

23. Check your understanding • Consider the graph below. State whether each of the statements about the object whose motion is represented by this graph is true or false.

24. Check your understanding (cont’d) • moving in the positive direction. • moving with a constant velocity. • moving with a negative velocity. • slowing down. • changing directions. • speeding up. • moving with a positive acceleration. • moving with a constant acceleration.