1-D Constant Acceleration Review Why constant acceleration? Constant acceleration equations. Example...
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Transcript of 1-D Constant Acceleration Review Why constant acceleration? Constant acceleration equations. Example...
1-D Constant Acceleration
• 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
• Velocity from acceleration
– v = vo + at
• Position from acceleration
– 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
a
t
t
v
vo
t
x
xo
This is supposed to be parabola!
Constant acceleration equations
• 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
• Solve
• Multiply by 2a
• Solve runway problem again
Constant acceleration equations
(Note special case a = 0)
Problem solving strategy
• 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 distanceEstimate 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
– What are principles?– What are equations? – Write down what’s known– Solve reaction time distance– Solve braking time– Solve braking time distance