motion with constant acceleration
DESCRIPTION
Motion with constant acceleration. Lecture deals with a very common type of motion: motion with constant acceleration After this lecture, you should know about: Kinematic equations Free fall. Summary of Concepts (from last lecture). kinematics: A description of motion - PowerPoint PPT PresentationTRANSCRIPT
04/19/23 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
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Motion with constant accelerationMotion with constant acceleration
Lecture deals with a very common type of motion: motion with constantacceleration
After this lecture, you should know about:
Kinematic equationsFree fall.
04/19/23 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
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Summary of ConceptsSummary of Concepts(from last lecture)(from last lecture)
kinematics: A description of motion position: your coordinates displacement: Δx = change of position distance: magnitude of displacement velocity: rate of change of position
average : Δx/Δtinstantaneous: slope of x vs. t
speed: magnitude of velocity acceleration: rate of change of velocity
average: Δv/Δtinstantaneous: slope of v vs. t
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Motion with constant acceleration in 1D Kinematic equations
An object moves with constant acceleration when the instantaneousacceleration at any point in a time interval is equal to the value of the average acceleration over the entire time interval.
Choose t0=0:
Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
04/19/23 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
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Motion with constant acceleration in 1D Kinematic equations (II)
Because velocity changes uniformly with time, the average velocity in the time interval is the arithmetic average of the initial and finalvelocities:
(1)
(2)
Putting (1) and (2) together:
04/19/23 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
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Motion with constant acceleration in 1D Kinematic equations (III)
The area under the graph of velocityvs time for a given time interval is equal to the displacement Δx of theobject in that time interval
04/19/23 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
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Motion with constant acceleration in 1D Kinematic equations (IV)
Putting the following two formulas together another way:
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Motion with constant acceleration in 1D
Kinematic equations (V)
Δx = v0t + 1/2 at2 (parabolic)
Δv = at (linear)
v2 = v02 + 2a Δx (independent of time) 0
5
10
15
20
0 5 10 15 20
v (m/s)
t (seconds)
0
50
100
150
200
0 5 10 15 20
x (meters)
t (seconds)
0
0.5
1
1.5
2
0 5 10 15 20
a (m/s2)
t (seconds)
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Use of Kinematic EquationsUse of Kinematic Equations
Shows velocity as a function of acceleration and time
Use when you don’t know or need the displacement
Gives displacement as a function of velocity and time
Use when you don’t know or need the acceleration
Gives displacement given time, velocity & acceleration
Use when you don’t know or need the final velocity
Gives velocity as a function of acceleration and displacement
Use when you don’t know or need the time
Example for motion with a=const in 1D: Example for motion with a=const in 1D: Free fallFree fall
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The Guinea and Feather tube
Earth’s gravity acceleratesobjects equally, regardless of their mass.
Experimental observations:
04/19/23 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
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Free Fall PrinciplesFree Fall Principles
Objects moving under the influence of gravity only are in free fallFree fall does not depend on the object’s original motion
Objects falling near earth’s surface due to gravity fall with constant acceleration, indicated by gg = 9.80 m/s2
g is always directed downward
» toward the center of the earth Ignoring air resistance and assuming g doesn’t vary with
altitude over short vertical distances, free fall is constantly accelerated motion
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Summary: Constant AccelerationSummary: Constant Acceleration
Constant Acceleration:
x = x0 + v0xt + 1/2 at2
vx = v0x + at
vx2 = v0x
2 + 2a(x - x0)
Free Fall: (a = -g)
y = y0 + v0yt - 1/2 gt2
vy = v0y - gt
vy2 = v0y
2 - 2g(y - y0)
x
yup
down
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A ball is thrown straight up in the air and returns to its initial position. For the time the ball is in the air, which of the following statements is true?1 - Both average acceleration and average velocity are zero.2 - Average acceleration is zero but average velocity is not zero.3 - Average velocity is zero but average acceleration is not zero.4 - Neither average acceleration nor average velocity are zero.
Example 1Example 1
correct
Free fall: acceleration is constant (-g)Initial position = final position: Δx=0
averaged vel = Δx/ Δt = 0
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Free Fall dropping & throwingFree Fall dropping & throwing
Drop Initial velocity is zeroAcceleration is always g = -9.80 m/s2
Throw Down Initial velocity is negativeAcceleration is always g = -9.80 m/s2
Throw Upward Initial velocity is positive Instantaneous velocity at maximum
height is 0Acceleration is always g = -9.80 m/s2
vo= 0 (drop)
vo< 0 (throw)
a = g
v = 0
a = g
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A ball is thrown downward (not dropped) from the top of a tower. After being released, its downward acceleration will be:
1. greater than g
2. exactly g
3. smaller than g
Throwing Down QuestionThrowing Down Question
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Example 2Example 2A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true: 1. velocity is zero and acceleration is zero2. velocity is not zero and acceleration is zero3. velocity is zero and acceleration is not zero4. velocity is not zero and acceleration is not zero
correct
Acceleration is the change in velocity. Just because the velocity is zero does not mean that it is not changing.
At the top of the path, the velocity of the ball is zero, but the acceleration is not zero. The velocity at the top is changing, and the acceleration is the rate at which velocity changes.
Acceleration is not zero since it is due to gravity and is always a downward-pointing vector.
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Dennis and Carmen are standing on the edge of a cliff. Dennis throws a basketball vertically upward, and at the same time Carmen throws a basketball vertically downward with the same initial speed. You are standing below the cliff observing this strange behavior. Whose ball is moving fastest when it hits the ground? 1. Dennis' ball 2. Carmen's ball 3. Same
vv00
vv00
DennisDennisCarmenCarmen
HH
vvAA vvBB
Example 3AExample 3A
Correct: v2 = v02 -2gΔy
On the dotted line:
Δy=0 ==> v2 = v02
v = ±v0
When Dennis’s ball returns
to dotted line its v = -v0
Same as Carmen’s
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Dennis and Carmen are standing on the edge of a cliff. Dennis throws a basketball vertically upward, and at the same time Carmen throws a basketball vertically downward with the same initial speed. You are standing below the cliff observing this strange behavior. Whose ball hits the ground at the base of the cliff first?
1. Dennis' ball 2. Carmen's ball 3. Same
Example 3BExample 3B
correct
vv00
vv00
DennisDennisCarmenCarmeny=y0
vvAA vvBB y=0
Time for Dennis’s ball to return to the dotted line:
v = v0 - g t
v = -v0
t = 2 v0 / g
This is the extra time taken by Dennis’s ball
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Example 4Example 4An object is dropped from rest. If it falls a distance D in time t then how far will if fall in a time 2t ?
1. D/4 2. D/2 3. D 4. 2D 5. 4D
Correct x=1/2 at2
Follow-up question: If the object has speed v at time t then what is the speed at time 2t ?
1. v/4 2. v/2 3. v 4. 2v 5. 4v
Correct v=at
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Which of the following statements is most nearly correct?
1 - A car travels around a circular track with constant velocity.2 - A car travels around a circular track with constant speed.3- Both statements are equally correct.
Example 5Example 5
correct
The direction of the velocity changes when going around circle.• Speed is the magnitude of velocity -- it does not have a
direction and therefore does not change
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Motion in 2DMotion in 2D
After this lecture, you should know about:Vectors.Displacement, velocity and acceleration in 2D.Projectile motion.
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One DimensionOne Dimension
Define origin Define sense of direction Position is a signed number (direction and magnitude) Displacement, velocity, acceleration are also specified by
signed numbers
} Reference Frame
0 1 2 3 4……-4 -1-2-3
VectorsVectors
There are quantities in physics which are determined uniquely by one number:
Mass is one of them.
Temperature is one of them.
Speed is one of them.
We call those scalars. There are others where you need more than one number; for
instance for 1D motion, velocity has a certain magnitude--
that's the speed--
but you also have to know whether it goes this way or that.
So there has to be a direction.
We call those vectors.
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04/19/23 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University
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Two DimensionsTwo Dimensions Again, select an origin Draw two mutually perpendicular lines meeting at the origin Select +/- directions for horizontal (x) and vertical (y) axes Any position in the plane is given by two signed numbers A vector points to this position
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Properties of vectorsProperties of vectors Equality of two Vectors
Two vectors are equal if they have the same magnitude and the same direction
Movement of vectors in a diagramAny vector can be moved parallel to itself without being affected
Negative VectorsOne vector is the negative of another one if they have both the same
magnitude but are 180° apart (opposite directions)
Resultant VectorThe resultant vector is the sum of a given set of vectors
Position can be anywhere in the plane
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Adding and subtracting vectors Adding and subtracting vectors geometricallygeometrically
R1
R2
D=R2-R1
R=R1+R2
D
y
x
Multiplying or Dividing a Vector by Multiplying or Dividing a Vector by a Scalara Scalar
The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided
by the scalar If the scalar is positive, the direction of the result is
the same as of the original vector If the scalar is negative, the direction of the result is
opposite that of the original vector
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Components of a VectorComponents of a Vector A component is a part It is useful to use rectangular components
These are the projections of the vector along the x- and y-axes
The x-component of a vector is the projection along the x-axis
The y-component of a vector is the projection along the y-axis
Then, one can define the component vectors
Attention: θ is measured counter-clock-wise with respect to the positive x-axis
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Components of a vector (II)Components of a vector (II)
The components are the legs of the right triangle whose hypotenuse is
May still have to find θ with respect to the positive x-axis
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Adding Vectors AlgebraicallyAdding Vectors Algebraically Choose a coordinate system and sketch the vectors Find the x- and y-components of all the vectors Add all the x-components
This gives Rx: Add all the y-components
This gives Ry: Use the Pythagorean Theorem to find the magnitude of the
resultant: Use the inverse tangent function to find the direction of R:
Inversion is not unique, the value will be correct only if the angle lies in the first or fourth quadrant
In the second or third quadrant, add 180°
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Example 6Example 6
Can a vector have a component bigger than its magnitude?YesNo
The square of magnitude of a vector is given in terms of its components by
R2= Rx 2+ Ry
2
Since the square is always positive the components cannot be larger than the magnitude
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Example 7Example 7 The sum of the two components of a non-zero
2-D vector is zero. Which of these directions is the vector pointing in?45o
90o
135o
180o
The sum of components is zero implies Rx = - Ry
The angle, θ = tan-1(Ry / Rx) = tan-1 -1 = 135o = -45o
(not unique, ± multiples of 2 θ)
-45o
135o
2D motion: Displacement2D motion: Displacement
The position of an object is described by its position vector,
The displacement of the object is defined as the change in its position
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2D motion: Velocity and acceleration2D motion: Velocity and acceleration The average velocity is the ratio of the displacement to the time interval
for the displacement
The instantaneous velocity is the limit of the average velocity as Δt approaches zeroThe direction of the instantaneous velocity is along a line that is
tangent to the path of the particle and in the direction of motion The average acceleration is defined as the rate at which the velocity
changes
The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero
Ways an object might accelerate: The magnitude of the velocity (the speed) can changeThe direction of the velocity can changeBoth the magnitude and the direction can change
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Kinematics in Two DimensionsKinematics in Two Dimensions
x = x0 + v0xt + 1/2 axt2
vx = v0x + axt
vx2 = v0x
2 + 2ax Δx
y = y0 + v0yt + 1/2 ayt2
vy = v0y + ayt
vy2 = v0y
2 + 2ay Δy
x and y motions are independent!They share a common time t
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2D motion: Projectile motion2D motion: Projectile motion
Dimensional Analysis:
Strategy:
Motion of a soccer ball
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Kinematics for Projectile MotionKinematics for Projectile Motion
aaxx = 0 a = 0 ayy = -g = -g
x = x0 + vxt
vx = v0x
y = y0 + v0yt - 1/2 gt2
vy = v0y - gt
vy2 = v0y
2 - 2g Δy
x and y motions are independent!They share a common time t
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Projectile MotionProjectile Motion
y ~ -x2, i.e. parabolic dependence on x
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Projectile Motion:Maximum height reached
Time taken for getting there
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Projectile Motion: Maximum Range
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Projectile Motion at Various Initial Projectile Motion at Various Initial AnglesAngles
Complementary values of the initial angle result in the same rangeThe heights will be
different
The maximum range occurs at a projection angle of 45o
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Soccer BallSoccer Ball
Check limiting cases
Make sense of what you get
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