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Announcements. No class next Monday (MLK day). Equations of Motion. Tractable cases. §2.5–2.6. Find Position from Velocity. Generally: velocity is slope of a position-time graph . Conversely, position is the area under a velocity-time graph. What is this when v is constant ?.

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Announcements

Announcements

  • No class nextMonday (MLK day)


Equations of motion

Equations of Motion

Tractable cases

§2.5–2.6


Find position from velocity

Find Position from Velocity

  • Generally: velocity is slope of a position-time graph.

  • Conversely, position is the area under a velocity-time graph.

  • What is this when v is constant?


Area under a v t graph

distance units

Area under a v-t graph

area = (a m/s)(b s) = ab m

speed (m/s)

a

b

time (s)


Constant velocity motion

Constant-Velocity Motion

  • v = Dx/Dt = constant throughout process

  • Dx = vDt

  • xf = xi + Dx = xi + vDt

  • Can also use this with average v


Find velocity from acceleration

Find Velocity from Acceleration

  • General case: acceleration is slope of a velocity-time graph.

  • Conversely, velocity is the area under an acceleration-time graph.

  • What is this when a is constant?


Constant acceleration motion

Constant-Acceleration Motion

  • Instantaneous accel = average accel

  • a = Dv/Dt

  • Dv = velocity change over time Dt

  • Dv = aDt

  • v = v0 + Dv = v0 + aDt


Acceleration on an x t graph

Acceleration on an x-t Graph

  • Velocity is the slope of a position-time graph

  • Acceleration means a changing slope

    • A constant slope means a straight x-t line

    • A varying slope means a curved x-t line

  • Positive acceleration = concave up

  • Negative acceleration = concave down


Visualize acceleration

Visualize Acceleration

Young and Freedman, Fig. 2.8

Board Work:

  • Signs of v

  • Signs of a


Acceleration

slope = velocity

d

t

slope = acceleration

v

area = distance

t

a

area = velocity

t

Acceleration

Starting from a traffic light that turns green


Equations of motion1

Equations of Motion

  • What are velocity and position under conditions of constant acceleration?


Formulas from constant x acceleration

Formulas from Constant x-Acceleration

  • Velocity change Dv = aDt

  • Velocity vt = v0 + Dv = v0 + aDt

  • Position change Dx = v0Dt + 1/2 a (Dt)2

  • Position xt = x0 + v0Dt + 1/2 a (Dt)2


Another form constant a

Another Form (constant a)

  • If you don’t know Dt and want v:

x = x0 + v0Dt + 1/2a (Dt)2Dt = Dv/a

x – x0 = v0 Dv/a + 1/2a (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 work?


Another form constant a1

Another Form (constant a)

  • If you don’t know a but know v, v0, and Dt:

x = x0 + v0Dt + 1/2a (Dt)2a = Dv/Dt = (v–v0)/Dt

x = x0 + v0 Dt + 1/2((v–v0)/Dt) (Dt)2

x – x0 = v0 Dt + 1/2v Dt – 1/2v0 Dt

x – x0 = v0 Dt – 1/2v0 Dt + 1/2v Dt

x – x0 = 1/2 (v0 + v)Dt

Do units work?


Example problem

Example Problem

A car 3.5 m in length traveling at 20 m/s approaches an intersection. The width of the intersection is 20 m. The light turns yellow when the front of the car is 50 m from the beginning of the intersection. If the driver steps on the brake, the car will slow at –3.8 m/s2 and if the car steps on the gas the car will accelerate at 2.3 m/s2. The light will be yellow for 3 s.

To avoid being in the intersection when the light turns red, should the driver use the brake or the gas?


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