1-D Constant Acceleration
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Review
Why constant acceleration?
Constant acceleration equations.
Example 1 - runway.
Problem solving strategy.
Example 2 - braking.
Falling Objects.
Example 3 – falling from tower
Example 4 – falling from tower.
Example 5 – throwing ball up.
Review
• Position
• Velocity
– vavg = (x2 –x1)/Δt = Δx/Δt
– vinst = limt->0(Δx/Δt ) = dx/dt
• Acceleration
– aavg = (v2 –v1)/Δt = Δv/Δt
– ainst = limt->0(Δv/Δt ) = dv/dt = d2x/dt2
• Signs (+/-) of x, v, and a must agree!
Position, Velocity, and Acceleration
• Car entering highway, then exiting
• Elevator going to upper floors, returning
• Car backing out of driveway
• Ball dropping
• Ball thrown up
• Must be sign consistent between x, v, a!
• Up positive, down negative (down positive)
Why constant acceleration?
• Position -> velocity -> acceleration – why not go
further?
– Constant acceleration problems common, changing
acceleration problems not!
– Newton’s 2nd Law involves acceleration.
• F=ma
– We deal with changing acceleration if we need to!
Constant acceleration equations
• Acceleration constant
a
t
• Velocity from acceleration
v
– v = vo + at
vo
• Position from acceleration
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x = vt (no! v is changing!)
x = (v + vo) / 2 * t
x = (at + vo + vo) / 2 * t
x = at2/2 + 2vot/2
x = ½ at2 + vot + xo
t
x
xo
This is supposed
to be parabola!
t
Constant acceleration equations
𝑎 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑣 = 𝑎𝑡 + 𝑣𝑜
1 2
𝑥 = 𝑎𝑡 + 𝑣𝑜 𝑡 + 𝑥𝑜
2
• Simplify for a=0 , v= 0
• (+/-) consistency
• Unit calculations
• We will provide on exam
Power rule of calculus
Example 1 - Runway
• Example 2.7 runway design
– Solve with time as intermediate step
• Find distance at takeoff velocity
• Find takeoff velocity at distance
– Solve without time as intermediate step
Solve without finding time
• Given
𝑣 = 𝑎𝑡 + 𝑣𝑜
𝑥=
1 2
𝑎𝑡 + 𝑣𝑜 𝑡 + 𝑥𝑜
2
• Solve
𝑡 = 𝑣 − 𝑣𝑜
1 𝑣 − 𝑣𝑜
𝑥= 𝑎
2
𝑎
2
+ 𝑣𝑜
𝑎
𝑣 − 𝑣𝑜
+ 𝑥𝑜
𝑎
• Multiply by 2a
2𝑎𝑥 = 𝑣 − 𝑣𝑜
2
+ 2𝑣𝑣𝑜 − 2𝑣𝑜 2 + 2𝑎𝑥𝑜
2𝑎 𝑥 − 𝑥𝑜 = 𝑣 2 − 𝑣𝑜 2
• Solve runway problem again
Constant acceleration equations
(Note special case a = 0)
Problem solving strategy
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Read problem well.
Decide what object you’re going to study.
Draw diagram, axes, positive/negative.
Decide principles that apply.
Write “knowns” and “unknowns”.
Decide relevant equations (validity?)
– OK to “just write equations, and see what happens” IF THEY ARE RELEVANT!
• What is “the trick” to the problem?
– Translating a qualitative question into numbers, criterion, or equation.
• Do calculation (round result)
• Does it make sense?
• Do units check?
Example 2 – Braking
• Example 2.9 braking distance
Estimate minimum stopping distance of car travelling at 50 km/h (14 m/s)
if maximum acceleration is -6 m/s2 and reaction time is 0.5 s
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What are principles?
What are equations?
Write down what’s known
Solve reaction time distance
Solve braking time
Solve braking time distance