Motion in One Dimension

1. Motion in One Dimension

2. Position • The object’s position is its location with respect to a chosen reference point. • Consider the point to be the origin of a coordinate system • In the diagram, allow the road sign to be the reference point

3. Motion = position changes with time • The position-time graph shows the motion of the particle (car). • Note the relationship between the position of the car and the points on the graph.

4. Displacement vs. Distance • Displacement: change in position during some time interval. • Represented as x x ≡xf- xi • SI units are meters (m) • x can be positive or negative • Distance: the length of a path followed by a particle; always positive.

5. Average Velocity • The average velocity is the rate at which displacement occurs • The x indicates motion along the x-axis • The dimensions are length / time [L/T] • The SI units are m/s. • It is also the slope of the line connecting initial and final points on the position – time graph.

6. Average Speed • Speed is a scalar quantity • same units as velocity • total distance / total time: • The average speed has no direction and is always expressed as a positive number. • Average velocity and average speed give no details about the trip described.

7. Position vs. Time Graph:Average and Instantaneous Velocity • The instantaneous velocity is the slope of the line tangentto the x vs. t curve. • This would be the green line • The slopes of the light blue lines represent the average velocities • note that as t gets smaller, they approach the green line

8. Instantaneous Speed • The instantaneous speed is the magnitude of the instantaneous velocity, therefore can not be negative. • The instantaneous speed has no direction associated with it.

9. Average vs. Instantaneous • Averagespeed= over a time interval • Average speed for the whole trip • Average speed between t=5s and t=10s • Slope of secant line on x(t) graph • Instantaneous speed = at this moment • Speed at t = 8s • Speed at the moment it passed by the police car • Slope of tangent line on x(t) graph

10. Particle Under Constant Velocity • instantaneous velocity at any instant during a time interval is the same as the average velocity during that time interval. • vx = vx, avg • The mathematical representation of this situation is the equation • Let ti = 0 and tf = t; the equation becomes: xf = xi + vx t

11. Particle Under Constant Velocity, Graph • The graph represents the motion of a particle under constant velocity. • The slope of the graph represents the constant velocity. • The y-intercept is xi

12. Particle Under Constant Acceleration

13. Average Acceleration • Acceleration is the rate of change of the velocity: • SI units are m/s² • In one dimension, positive and negative can be used to indicate direction.

14. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as t approaches 0 • The term acceleration will mean instantaneous acceleration • If average acceleration is wanted, the word average will be included

15. Acceleration: velocity – time graph • The slope on the velocity-time graph reveals the acceleration: • Slope of the green (tangent) line represents the instantaneous acceleration. • Slope of the blue (secant) line is the average acceleration. • Constant acceleration: graph is linear, average and instantaneous values are equal The area under the graph corresponds to the displacement of the particle

16. Acceleration: position – time graph • Graph curves towards time axis = particle slows down • Graph curves away from time axis = particle slows down • Graph is linear = particle moves at constant velocity, so acceleration is zero.

17. Acceleration and Velocity • When an object’s velocity and acceleration are in the same direction, the object is speeding up. • When an object’s velocity and acceleration are in opposite directions, the object is slowing down. • If possible, take initial velocity direction as positive. If not, don’t.

18. Acceleration and Velocity • Images are equally spaced. The car is moving with constant positive velocity (shown by red arrows maintaining the same size) • Acceleration equals zero

19. Acceleration and Velocity • Velocity and acceleration are in the same direction • Acceleration is uniform (violet arrows, same length) • Velocity is increasing (red arrows are getting longer) • This shows positive acceleration and positive velocity

20. Acceleration and Velocity • Acceleration and velocity are in opposite directions • Acceleration is uniform • Velocity is decreasing (red arrows are getting shorter) • Positive velocity and negative acceleration

21. Acceleration and Velocity • In all the previous cases, the acceleration was constant • Shown by the violet arrows all maintaining the same length • The diagrams represent motion of a particle under constant acceleration.

22. Kinematic Equations – The Big Four

23. Kinematic Equations • The kinematic equations can be used with any particle under uniform acceleration. • You may need to use two (or more) of the equations to solve one problem. • x, Dx, v, a  can be either positive or negative! • Many times there is more than one correct way to solve a problem.

24. Kinematic Equations • For constant acceleration: • Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration. • Assumes ti = 0 and tf = t • Does not give any information about displacement.

25. Kinematic Equations, specific • For constant acceleration: • The average velocity can be expressed as the arithmetic mean of the initial and final velocities.

26. Kinematic Equations, specific • For constant acceleration: • This gives you the position of the particle in terms of time and velocities. • Doesn’t give you the acceleration.

27. Kinematic Equations, specific • For constant acceleration: • Gives final position in terms of velocity and acceleration. • Doesn’t tell you about final velocity.

28. Kinematic Equations, specific • For constant acceleration: • Gives final velocity in terms of acceleration and displacement. • Does not give any information about the time.

29. When a = 0 • When the acceleration is zero, • vxf = vxi = vx • xf = xi + vx t • i.e., constant acceleration includes constant velocity as a special case.

30. Example • A train starts from rest and moves with constant acceleration. It reaches 30 m/s the moment it has traveled 150 m. Calculate: • The acceleration of the train, and the time it takes to travel the first 150 m; • The time it takes, and the distance it travels between reaching 30 m/s until it reaches 50 m/s.

31. Free Fall

32. Galileo Galilei • 1564 – 1642 • Italian physicist and astronomer. • Formulated laws of motion for objects in free fall. • Supported heliocentric universe.

33. Freely Falling Objects • A freely falling object is moving under the influence of gravity alone. • We will neglect air resistance. • Acceleration does not depend on the launching conditions of the object: • Dropped – released from rest • Launched downward • Launched upward

34. Acceleration of Freely Falling Object • The acceleration of an object in free fall is directed downward, regardless of the initial motion. • Its magnitude is constant, g = 9.80 m/s2 • g decreases with altitude • g varies with latitude, Earth crust composition, etc • 9.80 m/s2 is the average at the Earth’s surface • It doesn’t come with a negative sign included

35. Acceleration of Free Fall • Free fall motion is constantly accelerated motion in one dimension. • Let upward be positive • Use: • kinematic equations • with ay = -g = - 9.80 m/s2

36. Kinematic Equations – The Big Four

37. Free Fall: dropped object • Initial velocity is zero • Let up be positive • Use the kinematic equations • use y instead of x • Acceleration is • ay = -g = -9.80 m/s2 vo= 0 a = - g

38. Free Fall: object launched downward • ay = -g = -9.80 m/s2 • Initial velocity vy,i< 0 • With upward being positive, initial velocity will be negative vo≠ 0 a = -g

39. Free Fall: object launched upward • Initial velocity is upward, so positive: vy,i> 0 • The instantaneous velocity at the maximum height is zero. • ay = -g = -9.80 m/s2 everywhere along the path, even at maximum height. v = 0 vo≠ 0 a = -g

40. Object Launched Upward. • If the motion is symmetrical, i.e. end point coincides with start point: • Then tup = tdown • Then vfinal = -vinitial • If the motion is not symmetrical: • Break the motion into various parts • Generally up and down

41. Free Fall Example • (Initial) velocity at A is …. and acceleration is …. • At B, the velocity is …. and the acceleration is …. • At C, the velocity is …. and the acceleration is …. • The total displacement is ….

42. Free Fall Example • Initial velocity at A is upward (+) and acceleration is -g (-9.8 m/s2) • At B, the velocity is 0 and the acceleration is -g (-9.8 m/s2) • At C, the velocity has the same magnitude as at A, but is in the opposite direction • The displacement is –50.0 m (it ends up 50.0 m below its starting point)