Announcements No class next Monday (MLK day) Do MasteringPhysics homework Homework 00 open ‘til Friday 10 PM Homework 01 will open soon Written Homework due next Friday Book problems 2.37 and 2.71

Describing Motion acceleration §2.3–2.4

Board Work A car waits at a stop light for 5 seconds, smoothly accelerates to 15 m/s over 5 seconds, and then continues at 15 m/s. Describe the car’s motion using a velocity-time graph.

Acceleration Rate of changing velocity Dv average acceleration = Dt over the entire interval Dv Dt at one instant instantaneous acceleration = lim = dv/dt = d2x/dt2

Poll Question What is the SI unit for acceleration? m. s. m·s. m/s.

Visualize Acceleration Board Work: Signs of v Signs of a Young and Freedman, Fig. 2.8

Group Work In earlier car scenario: What is the car’s acceleration during the different segments of its motion? Describe the car’s motion using an acceleration-time graph.

Equations of Motion as functions of time

Find Position from Velocity Generally: velocity is derivative of position wrt time dx/dt. Conversely, position is the integral of velocity over time. v = dx/dt dx = v dt ∫dx = ∫v dt x = ∫v dt + x0 What is this when v is constant?

Integral of v-t graph area = (a m/s)(b s) = ab m speed (m/s) distance units speed (m/s) a b time (s)

Constant Acceleration Instantaneous accel = average accel a = Dv/Dt Dv = velocity change over time Dt Dv = a Dt v = v0 + Dv = v0 + a Dt

Find Velocity from Acceleration General case: acceleration is derivative of velocity wrt time dv/dt. Conversely, velocity is the integral of acceleration over time. a = dv/dt dv = a dt ∫dv = ∫a dt v = ∫a dt + v0 What is this when a is constant?

Equations of Motion What are velocity and position under conditions of constant acceleration?

Formulas from Constant x-Acceleration Velocity change Dv = a Dt Velocity vt = v0 + Dv = v0 + a Dt Position change Dx = v0 Dt + 1/2 a (Dt)2 Position xt = x0 + v0 Dt + 1/2 a (Dt)2

Another Form (constant a) If you don’t know Dt and want v: x = x0 + v0Dt + 1/2 a (Dt)2 Dt = Dv/a x – x0 = v0 Dv/a + 1/2 a (Dv/a)2 2a (x–x0) = 2v0 (v–v0) + (v–v0)2 2a (x–x0) = 2vv0 – 2v02 + v2 – 2vv0 + v02 2a (x–x0) = 2vv0 – 2vv0 + v2 + v02 – 2v02 2a (x–x0) = v2 – v02 v2 = v02 + 2a (x–x0) Do units check out?

Another Form (constant a) If you don’t know a but know v, v0, and Dt: x = x0 + v0Dt + 1/2 a (Dt)2 a = Dv/Dt = (v–v0)/Dt x = x0 + v0 Dt + 1/2 ((v–v0)/Dt) (Dt)2 x – x0 = v0 Dt + 1/2 v Dt – 1/2 v0 Dt x – x0 = v0 Dt – 1/2 v0 Dt + 1/2 v Dt x – x0 = 1/2 (v0 + v) Dt Do units check out?