PPT-Rectilinear Motion.pptx

application of calculus

1. Review OF THE PREVIOUS LESSON

2. What was the previous discussion last meeting all about?

3. Rate of change of a function It is defined as the rate at which one quantity is changing with respect to another quantity.

4. Average rate of change of a function

5. Instantaneous rate of change

6. Activity

7. Guide questions What was the video all about? What was the video all about? What was the video all about? 2. Describe the motion of the vehicle in the video. 3. What can be measured in a moving object?

8. Rectilinear motion

9. objectives

10. Kinematics Kinematics is the study of motion of a system of bodies without directly considering the forces or potential fields affecting the motion. In other words, kinematics examines how the momentum and energy are shared among interacting bodies.

11. Rectilinear motion Rectilinear motion is another name for straight-line motion (vertical or horizontal orientation). This type of motion describes the movement of a particle or a body.

12. Guide questions 1. Does the video showcase a rectilinear motion? 2. What are some examples of rectilinear motion?

13. Quantities Scalar Quantities Vector Quantities Distance Displacement Speed Velocity Time Acceleration

14. Scalar quantities Scalar quantity is defined as the physical quantity with magnitude and no direction. Example: Distance, Speed, Time

15. Review of concepts in physics

16. Scalar quantities Time is the progression of events from the past to the present into the future. Units: Seconds ,Minutes, Hours, Day, Week, Month

17. Scalar quantities Distance is a scalar quantity that refers to "how much ground an object or a particle has covered" during its motion. Units: Meter, Kilometer, Miles, etc.

18. Scalar quantities Speed is a scalar quantity that refers to "how fast an object is moving." Speed can be thought of as the rate at which an object covers distance.   Units: m/s, km/hr., mi/s, mi./min, etc.

19. Vector quantities Vector quantity is defined as the physical quantity that has both magnitude and direction. Example: Velocity, Acceleration

20. Vector quantities Velocity is a vector quantity that refers to "the rate at which an object changes its position."

21. Vector quantities The instantaneous velocity is the specific rate of change of position (or displacement) with respect to time at a single point.

22. Vector quantities Average velocity is the average rate of change of position (or displacement) with respect to time over an interval.

23. Vector quantities Formula of Velocity   Units: m/s, km/hr., mi/s, mi./min, etc.

24. Vector quantities Acceleration is a vector quantity that is defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.   Units:  

25. Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   The average velocity of P at time to is .   The average velocity of a particle in a rectilinear motion is the average rate of change of its position function. Relationship of position/distance, Speed velocity and acceleration of a particle in a rectilinear motion

26. Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   ii. The instantaneous velocity of P at time t is .   The instantaneous velocity of a particle in a rectilinear motion is the instantaneous rate of change of its position function. The first derivative of the position function is the instantaneous velocity of P. Relationship of position/distance, Speed velocity and acceleration of a particle in a rectilinear motion

27. Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   iii. The speed of P at time t is .   T he speed of P is the absolute value of first derivative of the position function which is the velocity of P. Relationship of position/distance, Speed velocity and acceleration of a particle in a rectilinear motion

28. Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   iv. The acceleration of P at time t is .   The acceleration of P is the second derivative of the position function . It is also the first derivative of the velocity function of P. Relationship of position/distance, Speed velocity and acceleration of a particle in a rectilinear motion

29. Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   A verage velocity In stantaneous velocity S peed Acceleration   Relationship of position/distance, Speed velocity and acceleration of a particle in a rectilinear motion

30. Remark : If an object is acted on only by the force of gravity, and if time , the height of the object is, initial height of the object, meters above the ground and the velocity is, the initial velocity of object, meters per second, then the object’s height in meters above the ground at time t seconds is given by .   Relationship of position/distance, Speed velocity and acceleration of a particle in a rectilinear motion

31. Example A particle moves along a line so that its position at any time is given by the function where s is measured in meters and t is measured in seconds. Find the average velocity during the first 5 seconds. Find the distance traveled by the particle after 7 seconds. Find the instantaneous velocity when . Find the acceleration of the particle when .  

32. Relationship of position/distance, velocity and acceleration of a particle in a rectilinear motion Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   A verage velocity In stantaneous velocity . Speed Acceleration .  

33. Solution To find the average velocity of the particle during first 5 seconds, we use the formula at and .  To find the value of and , substitute and to the given function   when when

34. Solution To find the average velocity, substitute , , and to the formula: meters per second  

35. Relationship of position/distance, velocity and acceleration of a particle in a rectilinear motion Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   A verage velocity In stantaneous velocity . S peed Acceleration .  

36. Solution b. To find the distance traveled by the particle after 7 seconds, substitute to the position function. when  

37. Solution c. To find the instantaneous velocity of the particle when , we differentiate with respect to time  

38. Solution Substitute to the velocity function, meters per second  

39. Relationship of position/distance, velocity and acceleration of a particle in a rectilinear motion Let the position of a particle (P) (or position) on a straight line at time be . Note that is the position function.   A verage velocity In stantaneous velocity . S peed Acceleration .  

40. Solution d. To find the acceleration of the particle when , compute the second derivative of the position function or compute the first derivative of the velocity function .  

41. Solution Substitute to the acceleration function, meters per second squared  

42. Examples Example 2. Suppose a one-peso coin is dropped from a height of 100 meters, its height s at time t is where s is measured in meters and t is measured in seconds. Find the average velocity on the interval . Find the instantaneous velocity when . How long will it take the object to hit the ground? Find the velocity of the object when it hits the ground.  

43. Free body diagram

44. Answer Key meters per second meters per second  

45. Group ACTIVITY

46. Instruction The class will be divided into four groups. Each group will be given a problem to solve. The group will write their answers in a Manila Paper. After ten minutes, each group will choose a representative to present their outputs. Additionally, a rubric will be provided for each group.

47. Group 1 & 3 The height s at time t of a one-peso coin dropped from the top of a building is given by where s is measured in meters and t is measured in seconds. Find the average velocity on the interval . Find the instantaneous velocity when t=1 and t=2. How long will it take for one-peso coin hit the ground? Find the velocity of the one-peso coin when it hits the ground.