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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?
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?