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1D MotionSept. 2013

X1Contents1D Motion, Kinematics and Dynamics - definitionSpeed and VelocityAccelerationEquation of KinematicsFreely falling bodiesGraphical analysis of velocity and accelration2Solution 1

Check Equation

72Example 1 cont.After 3.00 sec of free-fall, what is the velocity v of the stone?Check Equation

73DisplacementTo describe the motion of an object, we must be able to specify the location of the object at all times

Displacement of the car is a vector drawn from the initial position to the final position. Displacement is a vector quantity

InitialFinal

It is important to note that the change in any variable is always the final value minus the initial value5Term - DisplacementDisplacement in one direction along the line is assigned a positive value, and a displacement in the opposite direction is assigned a negative valueDisplacement does not give information of the location, it gives information about the change in location in specific time.

(The answer is given at the end of the book.)1.A honeybee leaves the hive and travels a total distance of 2 km before returning to thehive. What is the magnitude of the displacement vector of the bee?6Average SpeedUsain Bolt run 100 meters in 9.77 seconds last August in Moscow. How fast did he run?

100 m / 9.77 sec = 10.23 m/sec

Average speed is the distance traveled divided by the time required to cover the distance (always positive)

7ExampleMy dog chased a cat in our backyard. It run a distance of 30 meters till the cat was lost. The average speed was 6 m/s. How much time did it take till the cat was gone?

8Example (cont.)

= 30 m / 6 (m/sec) = 5 secThe dog ran 5 sec till it lost the cat.

9Average VelocitySpeed indicates how fast an object is moving. However, speed does not reveal anything about the direction of the motion.

To describe both how fast an object moves and the direction of its motion, we need the vector concept of velocity.10Term Average Velocityis the initial location at timeis the final location at time

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Average Velocity (cont.)Average velocity is a vector that points in the same direction as the displacementwe can represent the direction of motion with a +/ sign

Objects A and B have thesame speed s = |v| = +10 m/s, but they have different velocities12

We can rewrite the equation as follows:We can see that the location is defined by the initial location summed up with the multiplication of the average velocity by the time period13ExampleA worm is located 2cm to the right of the origin when measurement start.It crawls towards the origin at constant velocity of 3 cm/secWrite a function that describe the relation between location and timeDraw a graph of the location at different timesDraw a graph of the average velocity at different timesWhat is the displacement during 10 sec of movementWhat is the distance passed in 10 secWhat is the displacement during the 4th second of movement

X=X0+Vt

V=3 cm/s14

Write a function that describe the relation between location and time

X=X0+VtV=-3 cm/sX(cm)First we decide of the positive direction it will be to the right

The positive direction impacts the signs of the velocity and the initial location, in this example the velocity will be negative and the location positive

X0=+2 cm15

X=X0+VtV=-3 cm/sX(cm)X0=+2 cmX=X0+VtX=2-3tWrite a function that describe the relation between location and time

= 0

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Draw a graph of the location at different times

X=X0+VtV=-3 cm/sX(cm)X0=+2 cmX=2-3t2X(cm)t(s)17

Draw a graph of the average velocity at different times

X=X0+VtV=-3 cm/sX(cm)X0=+2 cmV(cm/s)t(s)-318

What is the displacement during 10 sec of movement?X=X0+VtV=-3 cm/sX(cm)X0=+2 cmX=-310=-30cmX=X0+VtX-X0=VtX=Vt19

What is the distance passed in 10 sec?X=X0+VtV=-3 cm/sX(cm)X0=+2 cmS=|X|=|-30|=30cmWhen a body is moving at constant velocity at all times, the distance passed is equal to the absolute value of the displacement20

What is the displacement during the 4th second of movement?

X=X0+VtV=-3 cm/sX(cm)X0=+2 cmThe 4th second of movement, is 1 second long as any other second.

Constant velocity motion has constant displacement at any second

-31=-3cm=X=Vt21Instantaneous VelocityThe magnitude of average velocity is an average, hence does not convey any information about how fast you were moving or the direction of the motion at any instant during the trip

The instantaneous velocity of the car indicates how fast the car moves and the direction of the motion at each instant of timeThe magnitude of the instantaneous velocity is called the instantaneous speedBoth can change from one instant to another. Surely there were times when your car traveled faster than 20 m/s and times when it traveled more slowly.22Instantaneous Velocity (cont.)

23Instantaneous Velocity (cont.)The notation means that the ratio

is defined by a limiting process. Smaller and smaller values of t are used, so small that they approach zero.

As smaller t is used, x also becomes smaller.

However, the ratio does not become zero,

it approaches the value of the instantaneous velocity.

For brevity, we will use the word velocity to mean instantaneous velocity and speed to mean instantaneous speed.24

25Exercises p. 52

26Exercises p. 52

27

28AccelerationIn a wide range of motions, the velocity changes from moment to moment. To describe the manner in which it changes, the concept of acceleration is neededThe change in velocity may occur over a short or a long time interval

The velocity of a moving object may change in a number of ways. For example, it may increase, as it does when the driver of a car steps on the gas pedal to pass the car ahead. Or it may decrease, as it does when the driver applies the brakes to stop at a red light

The average acceleration is a vector that points in the same direction as delta V, the change in the velocity

29Acceleration (cont.)If the velocity is changing, then there is non-zero acceleration

30

Rewriting the equation:We received a new function that describes the velocity of a body at any time t, that started off with velocity Vo and travels at constant acceleration a during the period of time t.31

The v vs. t graph slope is the accelerationGraph of motion in constant accelerationV(m/s)t(s)V0Vt

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Negative slope represents negative accelerationAcceleration can be Positive or NegativeV(m/s)t(s)V0Vt33

Is it true to state that any time a body is decelerating, it is also slowing down?Whenever the acceleration and velocity vectors have opposite directions, the object slows down and is said to be decelerating34

No!35

Negative slope, represent negative accelerationV(m/s)t(s)-V0VtMotion is in negative direction, hence velocity is negative.

+XAttention!!! The magnitude of the velocity is increasing36

Is it true to state that any time a body have positive accelerating, it is also speeding up?37

No!38

V(m/s)t(s)-V0VtVelocity magnitude is decreasing, acceleration is positive

+XPositive SlopeVelocity is negative because motion is towards negative X, and in the same time the velocity is decreasing

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If velocity is increased while moving in positive direction, both velocity and acceleration are positiveV(m/s)t(s)V0Vt

+X40

If velocity is increased while moving in negative direction, both velocity and acceleration are negative+X

a

Negative Slope => Negative accelerationV(m/s)t(s)V0VtVelocity increased to the leftConst. Acceleration left41

If velocity is decreased while moving in positive direction, the velocity is positive, and the acceleration is negativet(s)V(m/s)V0Vt

+X

Velocity decreased to the rightConst. Acceleration to left42

If velocity is decreased while moving in negative direction, the velocity is negative, and the acceleration is positiveV(m/s)t(s)V0Vt+X

Va

Velocity decreased to the left

Const. Acceleration to right43

t(s)V(m/s)V0Vt

+XVelocity decreaseVelocity increaseConst. negative accelerationTemp stop to switch motion directionVelocity magnitude decreaseVelocity magnitude increase44

t(s)V(m/s)-V0

+XConst. positive accelerationVelocity decreaseVelocity IncreaseTemp stop to switch motion directionVelocity magnitude decreaseVelocity magnitude increaseVt45Instantaneous Acceleration is an objects acceleration at a particular instant of time

Instantaneous acceleration is a limiting case of the averageacceleration.

When the time interval becomes extremely small (approaching zero in the limit),

the average acceleration and the instantaneous acceleration become equal

Instantaneous Acceleration46

Constant acceleration in 1D

47

Constant acceleration in 1D

48

Constant acceleration in 1D

49Example 1The plane in the figure starts from rest (Vo = 0 m/s) when t0 = 0 s. The plane accelerates down the runway and at t=29 s attains a velocity of V=260 km/h, where the plus sign indicates that the velocity points to the right. Determine the average acceleration of the plane?

50Solution 1The average acceleration of the plane is defined as the change in its velocity divided by the elapsed time

Assuming the acceleration of the plane is constant, a value of nine kilometers per hour per second means the velocity changes by +9.0 km/h during each second of the motion

51Solution 1- cont.It is customary to express the units for acceleration solely in terms of SI units.convert the velocity units from km/h to m/s :

The average acceleration then becomes

The velocity changes by 2.5 m/s during each second of the motion

52Example 2A drag racer crosses the finish line, and the driver deploys a parachute and applies the brakes to slow down. The driver begins slowing down when t0=9.0 s and thecars velocity is V0=28 m/s. When t=12.0 s, the velocity has been reduced to V=13 m/s. What is the average acceleration of the dragster?

53Solution 2The average acceleration of an object is always specified as its change in velocity,V-V0 , divided by the elapsed time, t-t0. This is true whether the final velocity is less than the initial velocity or greater than the initial velocity

54Solution 2 cont.

55

Exercises p. 54

56Excersises

57Equation of Kinematics (a = const.)We already saw earlier the equation of velocity:

V = V0 + at

If X0 = 0 and t0 = 0 we reduce the average velocity equation to:

58Equation of Kinematics (a = const.)Because the acceleration is constant, the velocity increases at a constant rate. Thus, the average velocity is midway between the initial and final velocities

And we get the following functions:

59Equation of Kinematics (a = const.)If t0 = 0 then we get

Arranging the equation:

And finally:

60Summary of equations

61Example

62SolutionUsing equation:

Since the spacecraft is slowing down, the acceleration must be opposite to the velocity

Both of these answers correspond to the same displacement (x= +215 km), but each arises in adifferent part of the motion

63Solution cont.

Sometimes there are two possible answers to a kinematics problem, each answercorresponding to a different situation64Exercises

65Exercises

66Exercises

67Freely falling bodiesIts well known that gravity causes objects to fall downward.In the absence of air resistance, all bodies at the same location above the earth fall vertically with the same acceleration. If the distance of the fall is small compared to the radius of the earth, the acceleration remains essentially constant.This idealized motion, in which air resistance is neglected and the acceleration is nearly constant, is known as free-fall68Free fall on the moonHere's the famous footage of the Apollo 15 astronaut that dropped a hammer & feather on the moon to prove Galileo's theory that in the absence of atmosphere, objects will fall at the same rate regardless of massThe acceleration due to gravity near the surface of the moon is approximately one-sixth as large as that on the earth

69Term Acceleration due to gravityThe acceleration of a freely falling body is called the acceleration due to gravity.Its magnitude is denoted by the symbol g. Its directed downward, toward the center of the earth. Near the earths surface, g is approximately

When the equations of kinematics are applied to free-fall motion, it is natural to use the symbol y for the displacement, since the motion occurs in the vertical or y direction

In reality, however,gdecreases with increasing altitude and varies slightly with latitude70 1ExampleA stone is dropped from rest from the top of a tall building.After 3.00 sec of free-fall, whatis the displacement y of the stone?

71Important!The acceleration due to gravity is always a downward-pointing vector. It describes how the speed increases for an object that is falling freely downward.

This same acceleration also describes how the speed decreases for an object moving upward under the influence of gravity alone, in which case the object eventually comes to a momentary halt and then falls back to earth74Example 2A football game customarily begins with a coin toss to determine who kicks off. The referee tosses the coin up with an initial speed of 5.00 m/s. In the absence of air resistance,how high does the coin go above its point of release?

75Solution 2

Upward initial velocity, but the acceleration due to gravity points downward.

Since the velocity and acceleration point in opposite directions, the coin slows down as it moves upward.

Eventually, the velocity of the coin becomes v=0 m/s at the highest point76Solution cont.

Check Equation

77Example 2 cont.What is the total time the coin is in the air before returning to its release point?Check Equation

78Example 3

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Solution 3The motion of an object that is thrown upward and eventually returns to earth has a symmetry that is useful to keep in mind from the point of view of problem solving. The calculations just completed indicate that a time symmetry exists in free-fall motion, in the sense that the time required for the object to reach maximum height equals the time for it to return to its starting point (T1 = T2).A type of symmetry involving the speed also exists.T1T280Solution 3 cont.If the pellet was shoot straight up with velocity +30 m/sec, than from symmetry we can tell that when it will be back at the initial height , it will have same velocity but in opposite direction.Hence the pellet will hit the ground in the same velocity on both cases.

81Exercises pg. 55

82Exercises p.56

83Graphical analysis of velocity and acceleration84SummaryTopics covered:Units and dimensionsUnit conversionLinear functions and graphTrigonometry

Next meeting:Scalars and vectors85The End86