# A Moving Frame of Reference - Wappingers Central School ... Describing Motion Chapter 3 What is a motion diagram? A Motion diagram is a useful tool to study the relative motion of objects. From motion diagrams, it is possible to observe an object under: Constant velocity Accelerating positively Accelerating negatively Or Stationary Motion Diagrams Constant Speed:

Positive Acceleration: Negative Acceleration: The Particle Model To simplify motion diagrams, we can concentrate all the motion through a single point at or near the center of gravity. The Particle Model Constant Speed: Positive Acceleration: Negative Acceleration: The Particle Model Constant Speed: Positive Acceleration:

Negative Acceleration: Determining Motion An objects motion can be determined if its initial and subsequent positions are identified relative to time. Initial Time Initial Position Initial Velocity = ti = di = vi Final Time Final Position Final Velocity = tf = df

= vf Average Velocity The average velocity is the ratio of displacement and time as follows: df - di d v= = t tf - t i (1) Where: d = the displacement vector t = change in time ti and di represent the starting position tf and df represent the final position

Average velocity does not tell you how the velocity varied during the time interval between the points, di and df. Graphical Representation of Velocity A graph of an object 16 14 12 Position (m) moving at constant velocity will consist of a straight line. The slope of this line will equal the average velocity of the object.

18 10 8 6 4 2 0 0 2 4 Time (s) 6 8 10

Average Acceleration An object in motion with changing velocity is under acceleration Acceleration is the rate of change of velocity as follows: vf - vi v a= = t tf - t i As with average velocity, the average (2) acceleration does not tell you how it varied during the time interval ti to tf. Graphical Representation of

Average Acceleration A graph of an object vf 16 14 12 Velocity (m/s) moving at constant acceleration will consist of a straight line. The slope of this line will equal the average acceleration of the object. The average between the initial and final values for velocity will equal the average.

18 10 vavg 8 6 4 vi 2 0 0 1 2 Tim e (s)

3 4 5 Finding Final Velocity Under Uniform Acceleration To find the final velocity when acceleration is uniform, all that is needed is the initial velocity, acceleration and time. By rearranging 2 to isolate vf, we obtain: vf = vi + at (3) An alternative method for calculating the final velocity is: vf2 = vi2 + 2ad

(4) Average Velocity during Uniform Acceleration For an object moving at constant acceleration, the average velocity is equal to the average of the initial plus final velocities. vi + vf vavg = 2 (5) Finding Displacement Under Uniform Acceleration When acceleration is uniform, the displacement depends on the objects acceleration, initial velocity and time. To find the displacement of an object during uniform acceleration, substitute 1 into 5 for vavg. vavg = d/t

vi + vf 2 vi + vf d/t = 2 vavg = d = 1 2 (vi + vf) t (1) (5) (6) Finding Displacement Under Uniform Acceleration

An alternative expression for (6) can be obtained by substituting 3 into 6: d = 1 2 (vi + vf) t vf = vi + at (3) d = 1 2 (vi + vi + at)t

d = 1 2 [2vi t + a(t)2] d = vi t + (6) 1 2 a(t)2 (7) Formulas for Motion of Objects Equations to use when an accelerating object

has an initial velocity. Form to use when accelerating object starts from rest (vi = 0). d = (vi + vf) t d = vf t vf = vi + at vf = at d = vi t + a(t)2 d = a(t)2 vf2 = vi2 + 2ad vf2 = 2ad

Formulas for Motion of Objects assuming d is displacement from origin and time starts at 0. Equations to use when an accelerating object has an initial velocity. Form to use when accelerating object starts from rest (vi = 0). d = (vi + vf) t = vavet d = vf t vf = vi + at vf = at

d = vi t + a(t)2 d = a(t)2 vf2 = vi2 + 2ad vf2 = 2ad