Physics Subject Area Test

1. Physics Subject Area Test MECHANICS: KINEMATICS

2. Motion in One DImension

3. To simplify the concept of motion, we will first consider motion that takes place in one direction. • One example is the motion of a commuter train on a straight track. • To measure motion, you must first choose a frame of reference. A frame of reference is a system for specifying the precise location of objects in space and time. • In the train example, any station along the route. One Dimensional Motion

4. Displacement is a change in position. • Displacement is not always equal to the distance traveled. • The SI unit of displacement is the meter, m. ∆ x = xf -xi Displacement – final position – initial position Displacement

5. Displacement is not always equal to the distance traveled. Example: If a gecko starts at an initial position of 20 cm and moves to the 80 cm mark, then retreats back to the 50 cm mark as its final position, How far has the gecko traveled? What is its displacement? The gecko traveled 90 cm, but its displacement is 30 cm.

6. Postive & Negative displacement • In general, right (east) is positive as well as upward (north) and left (west) is negative as well as downward (south).

7. Average velocity is the total displacement divided by the time interval during which the displacement occurred. In SI, the unit of velocity is meters per secondabbreviated as m/s. Average velocity

8. Example: • Consider a trip to a friend’s house 370 km to the west (negative direction) along a straight highway. If you left at 10 AM and arrived at 3 PM, what is your average velocity? This is your average. You did not travel at 74 km/h at every moment.

9. Velocity is not the same as speed. • Velocity describes motion with both direction and a numerical value (magnitude). • Speed has no direction, only magnitude. • Average speed is equal to the totaldistance traveled divided by the time interval. Velocity & speed

10. Consider an object whose position-time graph is not a straight line, but a curve. • We obtain different average velocities depending on the time interval. The instantaneous velocity is the velocity of the object at a specific point in the object’s path The instaneous velocity can be determined by measuring the slope of the line that is tangent to that point on the diatance-vs-time graph. Interpreting velocity graphically

11. acceleration

12. Acceleration – Rate at which velocity changes over time • An object accelerates if its speed, direction or both change. • Acceleration has directionand magnitude. • Acceleration is a vector quantity. Changes in velocity

13. Acceleration has the dimensions of lengthdivided by time squared. • SI units are m/s2 • Remember we have (m/s)/s = m/s2 Acceleration

14. Example: A bus slows down with an average acceleration of -1.8 m/s2. How long does it take the bus to slow down from 9.0 m/s to a complete stop?

15. Changes in Velocity • Consider a train moving to the right, so that the displacement and velocity are positive. • The slopeof the velocity-time graph is the average acceleration. • When the velocity in the positive direction is increasing, the acceleration is positive, as at A. • When the velocity is constant, there is no acceleration, as at B. • When the velocity in the positive direction is decreasing, the acceleration is negative, as at C.

16. Velocity & acceleration

17. When velocity changes by the same amount during each time interval, acceleration is constant. • The relationships between displacement, time, velocity, and constant acceleration are expressed by the equations shown on the next slide. Motion with constant acceleration

18. These equations apply to any object moving with constant acceleration. These equations use the following symbols:

19. Equations for constantly accelerated straight line motion

20. Example: • A racing car reaches a speed of 42 m/s. It begins a uniform negative acceleration , using its parachute and braking system, and comes to a complete rest 5.5 s later. Find the distance that the car travels during braking.

21. Vectors

22. A scalar is only a magnitude (length) (Example: Temperature, time, mass) • A vectorhas magnitude and direction (Example: displacement = 10 m East, Velocity= 50 mph west) Physical Quantities are of 2 types…

23. A vector will be symbolized by the “letter” with an arrow over it. The arrow indicates direction. Vectors are equal if they have the same units, magnitude, and direction. A vector can be moved anywhere parallel to itself.

24. To add vectors they must have the same units. • Tip-to -tail method put them head to tail and connect them so you end up with a triangle. • Parallelogram Method- (put them tail to tail) make vectors parallel and draw a line making 2 triangles Adding Vectors (attach)

26. Resultant Vector • The resultant vector is the sum of a given set of vectors More Properties of Vectors

27. Tip to tail- subtract by putting vector in the opposite direction • If you change the sign of a vector it is not the same vector. It is a new vector. • A – B does not equal B - A Subtracting Vectors

28. Components of a Vector • A component is a part • It is useful to use rectangular components • These are the projections of the vector along the x- and y-axes

29. The x-component of a vector is the projection along the x-axis • The y-component of a vector is the projection along the y-axis • Then, Components of a Vector, cont.

30. Useful Formulas.. The Pythagorean Theorem can only be used with right triangles!

31. When its not 900 R2= A2 + B2 – 2AB(COSӨ)

32. Find the magnitude of the sum of a 15 km displacement and a 25 km displacement when the angle between them is 900 and when the angle between them is 1350. Example 1

33. Find the horizontal and vertical components of the 100m displacement of a superhero who flies from the top of a tall building at an angle of 30.00 • (b) Suppose instead the superhero leaps in the other direction along a displacement vector B to the top of a flagpole where the displacement components are given Bx= -25m and BY=10.0m. Find the magnitude and direction of the displacement vector. Example Problem

34. A GPS receiver indicates that your home is 15.0 km and 400 north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home? • R= 12.39 • Ө= 158’ Example 2

35. Resolving a Vector Into Components +y The horizontal, or x-component, of A is found by Ax = A cosq. A Ay q Ax The vertical, or y-component, of A is found by Ay = A sin q. +x By the Pythagorean Theorem, Ax2 + Ay2 = A2 Every vector can be resolved using these formulas, such that Ais the magnitude of A, and qis the angle the vector makes with the x-axis. Each component must have the proper “sign” according to the quadrant the vector terminates in.

36. Analytical Method of Vector Addition 1. Find the x- and y-components of each vector. Ax = A cos q = Ay = A sin q = By = B sin q = Bx = B cos q = Cx = C cos q = Cy = C sin q = Rx= Ry = 2. Sum the x-components. This is the x-component of the resultant. 3. Sum the y-components. This is the y-component of the resultant. 4. Use the Pythagorean Theoremto find the magnitude of the resultant vector. Rx2 + Ry2 = R2

37. A roller coaster moves 215 ft horizontally and then rises 130 ft at an angle of 35.00 above the horizontally. Next, it travels 125 ft at an angle of 50.00 below the horizontal. Find the roller coaster’s displacement from its starting point to the end of this movement.

38. A quarter back takes the ball from the line of scrimmage, runs backwards for 15.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 60.0 yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

39. Vector Multiplication DOT PRODUCT scalar product A ∙ B A ∙ B = AB cosφ The product of the 2 vectors and the cosine of the angle between them

40. A ∙ B = (Axi + Ayj) (Bxi + Byj) • = AxBxi ∙ i + AxByi ∙ j + AyBxj ∙ i + AyByj ∙ j • i.i = j.j = k.k = 1 • and i.j = j.i = i.k = k.i = j.k = k.j = 0 • A ∙ B = AxBxi ∙ i + AyByj ∙ j • With 3 dimension: • A ∙ B = AxBxi ∙ i + AyByj ∙ j + AzBzk ∙ k

41. CROSS PRODUCT vector product A x B The product of the 2 vectors and the sine of the angle between them A x B is not the same as B x A … the direction is opposite

42. i x i = j x j = k x k = 0 i x j = k j x k = I k x i = j a x b = (a2b3 – a3b2 ) i + (a3b1 – a1b3) j + (a1b2- a2b1) k • k i j

43. Two vectors in component forms are written as : In evaluating the product, we make use of the fact that multiplication of the same unit vectors gives the value of 0, while multiplication of two different unit vectors result in remaining vector with appropriate sign. Finally, the vector product evaluates to vector terms :

44. Projectile Motion

45. Moving in the x and y direction • A projectile is an object shot through the air. This occurs in a parabola curve. A projectile is a motion in two dimensions

46. projectile- any object that moves through the air or through space, acted on only bygravity (and air resistance, if any) Object dropped Object thrown at an angle Object thrown up

47. The vertical acceleration of a projectile is caused by gravity, so ay = -9.8 m/s2 Parabolic Trajectory

48. g remains constant (g= -9.8m/s2) • a in the x direction is 0 because gravity is not acting on it. • Neglect air resistance • Neglect the effects of the earths rotation We assume that

49. Projectiles launched horizontally

50. To find how far the ball falls, you use the formula.y=viyt + 1/2gt2 1st second- 5m After 2 seconds- 20m After 3 seconds- 45m The curved path of a projectile produced is a parabola (caused by both horizontal motion and vertical motion. It must accelerate only in the vertical direction)