the 5 kinematic equations. the average speed is equal to the total distance traveled divided by the...
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The 5 Kinematic Equations
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The average speed is equal to the total distance traveled divided by the total time of travel
d v = ----t
During an acceleration the average speed is equal to midpoint between the initial and final velocity
vf + vi
2 v =
Find the average speed in the first 4 seconds Find the average speed in the
first 2 seconds
V(2) = ?
V(1) = ?
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We can rearrange these equations to create other useful equations..
Goal: Let’s derive an equation for the distance traveled by an accelerating object
v = ----t
d vf + vi
2 =
Now solve the right side for d: d = v t
Substitute v = (vf + vi ) / 2 d = (vf + vi ) t
2Now lets introduce
a = v/t = (Vf – Vi)/t
So t = (Vf – Vi)/a
which we can substitute above…..
Now we can expand out the parenthesis and derive equation #5 from this………..
d = (vf + vi ) (Vf – Vi)/a
2
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d = (vf + vi ) (Vf – Vi)
2 a
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Vf2 = Vi
2 + 2 a d This is a handy equation used to solve
problems like this:
A car accelerates from a speed of 20 m/s to 30 m/s at an acceleration of 2.0 m/s2.
How far will the car travel during this acceleration?
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How fast will an object dropped from rest be moving after 4 seconds?
How far will an object dropped from rest have fallen after 4 seconds?
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The rearrangement of equation #2………
a = v/t Solve this for v : v = at = Vf -Vi
Solve this for Vf:
vf = vi + at which is equation #3
Ex: A rocket moving at 1000 m/s accelerates at 20 m/s2 for 10 seconds to achieve a higher orbit. What will its new speed be after it reaches its new orbit?
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D
And that’s equation #4: d = vi t + ½ at2
Example: A car traveling at 10 m/s accelerates for 5.0 s at a rate of 2.5 m/s2. How far will it travel during those 5.0 seconds?
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Many of his experimental apparatus can be found at the Museo di Storia della Scienza in Florence, Italy
There we have the 5 Kinematic Equations. Galileo didn’t exactly write them algebraically like this, but he set up the ideas for measuring v, a, d and t and showed through reasoning, measurement proportions how v, d , a and t related to one another. Note there is no concept of force yet in Physics…that will come later with Isaac Newton.
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d = vi t + ½ at20
d t ½ a t2
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Distance vs Displacement5 meters east in 2 seconds
3 meters south in 1 second
The total distance d traveled is
The total displacement d is
Speed = d / t = v
Velocity= d / t = v
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Vector mathematics is different than scalar mathematics
Vectors have direction and must be drawn to scale and connected head-to-tail. They add up to a resultant.
Scalars are just numbers and can be added with regular math. They add up to a sum.
HEADTail
Start
Finish
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2 miles
4 miles 3 miles
Road runner travels 5 miles before Marvin sends him back. Road runner takes a shortcut back that saves him a mile. If the whole round trip took 1 hour, what is roadrunner’s
a) speed? b) displacement ?