Kinematics Horizontal and Vertical Equations in one dimension.
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Transcript of Kinematics Horizontal and Vertical Equations in one dimension.
KinematicsKinematics
Horizontal and Vertical Horizontal and Vertical EquationsEquations
in one dimensionin one dimension
Displacement Displacement →→ ΔΔx =x = x xff – x – xii
Velocity Velocity → v =→ v =
Acceleration → a =Acceleration → a =
Δx
Δt
ΔvΔt
So far…BASIC EQUATIONS LIST
We now know everything We now know everything we need to know to predict we need to know to predict
the path of an object in the path of an object in horizontal motionhorizontal motion
x t vx t vff v vii a a
If we know 3 of these, we can figure out the other 2.
atvv if
This equation can be rearranged to look like:
This equation is the 1st kinematics equation.
From before we know:
If we take the integral If we take the integral of this equation, we of this equation, we get :get :
Most of the time, xMost of the time, xi i will will be 0.be 0.
atvv if
atvv if
From this equation we can determine the 2nd equation.
2
2
1attvx i
This equation is the 2nd kinematics equation.
For the 3For the 3rdrd kinematics equation: kinematics equation:
vf = vi + a t t = (vf – vi) / a
x = vi t + ½ a t 2
x = vi [(vf – vi) / a] + a [(vf – vi) / a] 2
vf2 – vi
2 = 2 a x
vf2 = vi
2 + 2ax
Most of the time, xMost of the time, xii will be 0. will be 0.
Kinematics Formula SummaryKinematics Formula Summary
1.1. vvff = v = vii + a t + a t
2.2. x = vx = vii t + t + ½ a t ½ a t 22
3.3. vvff22 = v = vii
22 + 2 a + 2 a xx
For 1-D motion with constant acceleration:
To Remember…To Remember…
If there is acceleration, we need If there is acceleration, we need more than just more than just
x = vt x = vt• Because the object has either increased Because the object has either increased
speed, therefore going furtherspeed, therefore going further
• Or the object has decreased speed, Or the object has decreased speed, therefore not covering as much ground.therefore not covering as much ground.
When do you use each equation?When do you use each equation?
Identify what you want to know.Identify what you want to know. Identify the information given.Identify the information given. Eliminate equations based on this information.Eliminate equations based on this information. Example:Example:
A catcher catches a 90 mph fast ball. His glove compresses 4.5 cm. How long does it take to
come to a complete stop?
Looking for time
Given
90mph
4.5cm
We know that the ball is stopping
Example continuedExample continued
0.00224s or 2.24 ms Answer
Looking for time = t
Given
90mph = vi = 40.2m/s
4.5cm = x = 0.045m
stopping = vf = 0
vvff = v = vii + at + at
x = vx = viit + ½att + ½at22
vvff22 = v = vii
22 + 2ax + 2ax
Which equation do we use? Can we solve with just one?
Need to find acceleration first. Use #3. a = -17956m/s2
Then find time. Use #1.
You try:You try:
1.1. A dune buggy accelerates uniformly at A dune buggy accelerates uniformly at 1.5 m/s1.5 m/s22 from rest to 22 m/s. Find the from rest to 22 m/s. Find the total distance traveled and the total total distance traveled and the total time. time.
2.2. A car is moving at 30m/s when the A car is moving at 30m/s when the brakes are applied. It stops 2.5 s later. brakes are applied. It stops 2.5 s later. Find the car’s acceleration and how far it Find the car’s acceleration and how far it traveled in that time.traveled in that time.
x =161.3m
t = 14.7 s
a = -12m/s2
x = 37.5m
Galileo GalileiGalileo Galilei
1564 - 16421564 - 1642 In addition to In addition to
telescopes and his telescopes and his other pursuitsother pursuits
Galileo formulated Galileo formulated the laws that the laws that govern the motion govern the motion of objects in free of objects in free fallfall
Free FallFree Fall All objects moving under the influence of All objects moving under the influence of
gravity onlygravity only are said to be in free fall are said to be in free fall• Free fall does not depend on the object’s Free fall does not depend on the object’s
original motionoriginal motion All objects falling near the earth’s surface All objects falling near the earth’s surface
fall with a constant accelerationfall with a constant acceleration Gravity accelerates the object toward the Gravity accelerates the object toward the
earth the entire time it rises, and the earth the entire time it rises, and the entire time it falls.entire time it falls.
This acceleration is called “acceleration This acceleration is called “acceleration due to gravity,” and is indicated by “due to gravity,” and is indicated by “g”g”
Acceleration due to GravityAcceleration due to Gravity
gg = = 9.8 m/s²9.8 m/s²• When estimating, use When estimating, use gg 10 m/s10 m/s22
gg is always directed is always directed downwarddownward• Toward the center of the earthToward the center of the earth
Ignoring air resistance and assuming Ignoring air resistance and assuming gg doesn’t vary with altitude over doesn’t vary with altitude over short vertical distances, free fall is short vertical distances, free fall is constantly accelerated motionconstantly accelerated motion
Vertical Kinematics EquationsVertical Kinematics Equations
1.1. vvff = v = vii + gt + gt
2.2. y = vy = viit + ½gtt + ½gt22
3.3. vvff22 = v = vii
22 + 2gy + 2gy
Replace a with g
and x with y
Free Fall of a Free Fall of a droppeddropped object object
Initial velocity is Initial velocity is zerozero
Use the same general Use the same general kinematics equationskinematics equations• Generally use y instead of x Generally use y instead of x
since verticalsince vertical• Acceleration is Acceleration is gg = 9.8m/s = 9.8m/s22
Because the object is Because the object is speeding up, we will say speeding up, we will say that g is positivethat g is positive• This does not agree with This does not agree with
your book.your book.
vi = 0
a = g
Free Fall of Free Fall of an object an object thrownthrown downward downward
a = a = gg = 9.80 m/s = 9.80 m/s22
Initial velocity Initial velocity 0 0• g is positive because the g is positive because the
object is speeding up.object is speeding up.• We can choose that downward We can choose that downward
direction is positive when direction is positive when dealing with falling objects sodealing with falling objects so
• Initial velocity will be positive Initial velocity will be positive The only thing acting on the The only thing acting on the
object after launch is object after launch is gravity… FREE FALLgravity… FREE FALL
vi 0
a = g
Practice Problem:Practice Problem: You drop a ball from rest off a 120 You drop a ball from rest off a 120 m high cliff. Assuming air resistance is negligible, how m high cliff. Assuming air resistance is negligible, how long is the ball in the air? What is the ball’s speed long is the ball in the air? What is the ball’s speed when it strikes the ground at the base of the cliff?when it strikes the ground at the base of the cliff?
What’s your reaction time?What’s your reaction time?Using a partner, vertical kinematics equations Using a partner, vertical kinematics equations
and a ruler, determine your reaction time…and a ruler, determine your reaction time…
t = 4.95s
vf =48.5m/s
Average is 0.2s
The motion may be symmetricalThe motion may be symmetrical• Then tThen tupup = t = tdowndown
• Then the speed is the same at the same Then the speed is the same at the same heights, but direction is oppositeheights, but direction is opposite
The motion may not be symmetricalThe motion may not be symmetrical• Break the motion into various partsBreak the motion into various parts
Generally up and downGenerally up and down
Free Fall – Free Fall – an object an object thrown upwardthrown upward
Take each side separately… Take each side separately… Only have to solve one side…Only have to solve one side…
v = 0
For symmetrical situations…For symmetrical situations…
Up SideUp Side Initial velocity is Initial velocity is upward.upward. a = a = g g = -9.80 m/s= -9.80 m/s22 (slowing down)(slowing down) The instantaneous The instantaneous velocity at the velocity at the maximum height is maximum height is zerozero Final velocity is Final velocity is zero.zero.
Down SideDown Side Initial velocity is Initial velocity is
zero.zero. a = a = g g = 9.80 m/s= 9.80 m/s22
(speeding up)(speeding up) The instantaneous The instantaneous
velocity at the velocity at the maximum height is maximum height is
zerozero Final velocity is Final velocity is
downward.downward.
Time up = Time down
Non-symmetrical Non-symmetrical Free FallFree Fall
Need to divide the Need to divide the motion into motion into segmentssegments
Possibilities includePossibilities include• Upward and Upward and
downward portionsdownward portions• The symmetrical The symmetrical
portion back to the portion back to the release point and then release point and then the non-symmetrical the non-symmetrical portionportion
Combination MotionsCombination Motions
Multi-step ProblemsMulti-step ProblemsHow fast should you throw a ball straight How fast should you throw a ball straight
down from 40 m up so that its impact down from 40 m up so that its impact speed would be the same as a rock’s speed would be the same as a rock’s impact speed dropped from 60 m?impact speed dropped from 60 m?
19.8 m/sAnswer: