Motion with constant acceleration

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.


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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


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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:


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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


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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:


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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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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

04/19/23 26Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

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


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


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


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


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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Motion with constant acceleration

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