Describing Motion Chapter 3 What is a motion

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 object’s 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 - d i d v= = tf - t i t (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 moving at constant velocity will consist of a straight line. § The slope of this line will equal the average velocity of the object.

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= = tf - t i t § 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 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. vf vavg vi

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 + a. Dt (3) § An alternative method for calculating the final velocity is: vf 2 = vi 2 + 2 a. Dd (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 = Dd/Dt vi + vf 2 vi + vf Dd/Dt = 2 vavg = Dd = 1 2 (vi + vf) Dt (1) (5) (6)

Finding Displacement Under Uniform Acceleration § An alternative expression for (6) can be obtained by substituting 3 into 6: Dd = 1 2 (vi + vf) Dt vf = vi + a. Dt (3) Dd = 1 2 (vi + a. Dt) Dt Dd = 1 2 [2 vi Dt + a(Dt)2] Dd = v i Dt + 1 2 (6) a(Dt)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). Dd = ½ (vi + vf) Dt Dd = ½ v f Dt vf = vi + a. Dt vf = a. Dt Dd = vi Dt + ½ a(Dt)2 Dd = ½ a(Dt)2 vf 2 = vi 2 + 2 a. Dd vf 2 = 2 a. Dd

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 = ½ (vi + vf) t = vavet d = ½ vf t vf = vi + at vf = at d = vi t + ½ a(t)2 d = ½ a(t)2 vf 2 = vi 2 + 2 ad vf 2 = 2 ad