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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?

Area under a v-t graphsp

eed

(m

/s)

time (s)

area = (a m/s)(b s) = ab m

a

b

distance units

Constant-Velocity Motion

• v = x/t = constant throughout process

• x = vt

• xf = xi + x = xi + vt

• Can also use this with average v

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

• Instantaneous accel = average accel

• a = v/t

• v = velocity change over time t

• v = a t

• v = v0 + v = v0 + a t

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

Young and Freedman, Fig. 2.8

Board Work:2. Signs of v3. Signs of a

AccelerationStarting from a traffic light that turns green

d

t

v

t

a

t

area = velocity

area = distance

slope = velocity

slope = acceleration

Equations of Motion

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

Formulas from Constant x-Acceleration

• Velocity change v = a t

• Velocity vt = v0 + v = v0 + a t

• Position change x = v0 t + 1/2 a (t)2

• Position xt = x0 + v0 t + 1/2 a (t)2

Another Form (constant a)

• If you don’t know t and want v:

x = x0 + v0t + 1/2 a (t)2 t = v/a

x – x0 = v0 v/a + 1/2 a (v/a)2

2a (x–x0) = 2v0 (v–v0) + (v–v0)2

2a (x–x0) = 2vv0 – 2v02 + v2 – 2vv0 + v0

2

2a (x–x0) = 2vv0 – 2vv0 + v2 + v02 – 2v0

2

2a (x–x0) = v2 – v02

v2 = v02 + 2a (x–x0) Do units work?

Another Form (constant a)

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

x = x0 + v0t + 1/2 a (t)2

a = v/t = (v–v0)/t

x = x0 + v0 t + 1/2 ((v–v0)/t) (t)2

x – x0 = v0 t + 1/2 v t – 1/2 v0 t

x – x0 = v0 t – 1/2 v0 t + 1/2 v t

x – x0 = 1/2 (v0 + v) t Do units work?

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?