Lesson 1: Describing Motion with Words



Introduction to the Language of Kinematics

Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations.

Vectors, Scalars, Distance, Displacement, Speed, Velocity, Acceleration

Scalars and Vectors

• Scalars are quantities which are fully described by a magnitude alone.

• Vectors are quantities which are fully described by both a magnitude and a direction.

Distance and Displacement

• Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion.

• Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's change in position.

Describing Motion with Words

Speed and Velocity

Speed is a scalar quantity which refers to "how fast an object is moving." A fast-moving object has a high speed while a slow-moving object has a low speed. An object with no movement at all has a zero speed.

Velocity is a vector quantity which refers to "the rate at which an object changes its position." Imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. This would result in a zero velocity. Because the person always returns to the original position, the motion would never result in a change in position.

When evaluating the velocity of an object, you must keep track of its direction. For instance, you must describe an object's velocity as being 20 m/s, east.

Average Speed and Average Velocity

As an object moves, it often undergoes changes in speed. For example, during an average trip to school, there are many changes in speed. Rather than the speedometer maintaining a steady reading, the needle constantly moves up and down to reflect the stopping and starting and the accelerating and decelerating. At one instant, the car may be moving at 50 mi/hr and at another instant, it may be stopped (i.e., 0 mi/hr). Yet during the course of the trip to school the person might average a speed of 25 mi/hr.

The average speed during the course of a motion is often computed using the following equation:

[pic]

Meanwhile, the average velocity is often computed using the equation:

velocity (m/s) = displacement (m)

time (s)

Example

Anthony traveled a total distance of 1800 metres. His trip took 150 seconds. What was his average speed?

[pic]

Instantaneous Speed

Instantaneous Speed - speed at any given instant in time.

Average Speed - average of all instantaneous speeds; found simply by a distance/time ratio.

You might think of the instantaneous speed as the speed which the speedometer reads at any given instant in time and the average speed as the average of all the speedometer readings during the course of the trip.

Constant Speed

Moving objects don't always travel with erratic and changing speeds. Occasionally, an object will move at a steady rate with a constant speed. That is, the object will cover the same distance every regular interval of time. If the speed is constant, then the distance traveled every second is the same.

[pic]

Acceleration

Acceleration is a vector quantity which is defined as "the rate at which an object changes its velocity." An object is accelerating if it is changing its velocity.

Constant Acceleration

Sometimes an accelerating object will change its velocity by the same amount each second. This is known as a constant acceleration since the velocity is changing by the same amount each second.

Since accelerating objects are constantly changing their velocity, you can say that the distance traveled divided by the time taken to travel that distance is not a constant value. A falling object for instance usually accelerates as it falls. If you were to observe the motion of a free-falling object ; you would notice that the object averages a velocity of 5 m/s in the first second, 15 m/s in the second second, 25 m/s in the third second, 35 m/s in the fourth second, etc. Our free-falling object would be accelerating at a constant rate.

These numbers are summarized in the table below.

|Time Interval |Average Velocity During Time |Distance Traveled During Time |Total Distance Traveled from 0 s to End|

| |Interval |Interval |of Time Interval |

|0 - 1 s |5 m/s |5 m |5 m |

|1 - 2 s |15 m/s |15 m |20 m |

|2 - 3 |25 m/s |25 m |45 m |

|3 - 4 s |35 m/s |35 m |80 m |

For objects with a constant acceleration, the distance of travel is directly proportional to the square of the time of travel.

Calculating Acceleration

The acceleration of any object is calculated using the equation:

[pic]

Acceleration values are expressed in units of velocity per time. Typical acceleration units are m/s 2 .

Direction of the Acceleration Vector

Acceleration is a vector quantity so it will always have a direction associated with it. The direction of the acceleration vector depends on two factors:

• when the object is speeding up it is given the positive (+) direction

• when the object is slowing down it is given a negative (–) direction

1. A car is travelling at 30 m/s and takes 10 seconds to acceleration to a new speed of 35 m/s. What is its acceleration? 

2. A car accelerates from 8 m/s to 20 m/s, and takes 6 seconds to do it.   What is the acceleration?

Describing Motion with Diagrams

Ticker Tape Diagrams

A common way of analyzing the motion of objects in physics labs is to perform a ticker tape analysis. A long tape is attached to a moving object and threaded through a device that places a tick upon the tape at regular intervals of time – say 50 every second.

A large distance between dots indicates that the object was moving fast during that time interval. A small distance between dots means the object was moving slow during that time interval.

A changing distance between dots indicates a changing velocity and thus an acceleration. A constant distance between dots represents a constant velocity and therefore no acceleration.

Describing Motion with Position vs. Time Graphs

The Meaning of Slope for a p-t Graph

The slope of a position vs. time graph reveals velocity. For example, a small slope means a small velocity; a negative slope means a negative velocity; a constant slope (straight line) means a constant velocity; a changing slope (curved line) means a changing velocity. Thus the shape of the line on the graph (straight, curving, steeply sloped, mildly sloped, etc.) is descriptive of the object's motion.

Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The diagram below depicts such a motion.

The position-time graph would look like the graph at the right. Note that during the first 5 seconds, the line on the graph goes up 10 meters along the vertical (position) axis for every 1 second along the horizontal (time) axis. It has a slope of +10 meters/1 second (10 m/s) , the same as the velocity of the car.

Example 2

Consider a car moving with a constant velocity of +5 m/s for 5 seconds, stopping abruptly, and then remaining at rest (v = 0 m/s) for 5 seconds.

Note that for the first five seconds, the line on the graph goes up 5 meters along the vertical (position) axis for every 1 second along the horizontal (time) axis. Note also that during the last 5 seconds (5 to 10 seconds), the line goes up 0 meters. That is, the slope of the line is 0 m/s — the same as the velocity during this time interval.

Determining the Slope on a p-t Graph

The slope of the line on a position vs. time graph is equal to the velocity of the object. If the object is moving with a velocity of +4 m/s, then the slope of the line will be +4 m/s. If the object has a velocity of 0 m/s, then the slope of the line will be 0 m/s.

[pic]

The line slopes upwards to the right. But mathematically, by how much does it slope upwards along the vertical (position) axis per 1 second along the horizontal (time) axis? To answer this question use the slope equation:

[pic]

The calculations apply this method to determine the slope of the line in the graph above. Note that three different calculations are performed for three different sets of points on the line. In each case, the result is the same: the slope is 10 m/s.

So that was easy — rise over run is all that is involved.

Determine the velocity (i.e., slope) of the object as portrayed by the graph.

Describing Motion with Velocity vs. Time Graphs

The Meaning of Shape for a v-t Graph

Consider a car moving with a constant, rightward (+) velocity of +10 m/s. A car moving with a constant velocity is a car moving with zero acceleration.

[pic]

If the velocity-time data for such a car were graphed, the resulting graph would look like the graph at the right. Note that a motion with constant, positive velocity results in a line of zero slope (a horizontal line has zero slope) when plotted as a velocity-time graph. Furthermore, only positive velocity values are plotted, corresponding to a motion with positive velocity.

Now consider a car moving with a rightward (+), changing velocity – that is, a car that is moving rightward and speeding up or accelerating. Since the car is moving in the positive direction and speeding up, it is said to have a positive acceleration.

If the velocity-time data for such a car were graphed, the resulting graph would look like the graph at the right. Note that a motion with changing, positive velocity results in a diagonal line when plotted as a velocity-time graph. The slope of this line is positive, corresponding to the positive acceleration.

|Positive Velocity Zero Acceleration |Positive Velocity Positive Acceleration |

|[pic] |[pic] |

Acceleration vs. Deceleration

How can you tell if the object is speeding up (acceleration) or slowing down (deceleration)? Speeding up means that the velocity is increasing. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to -9 m/s is also speeding up.

[pic]

The Principle of Slope for a v-t Graph

If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration is negative, then the slope is negative (i.e., a downward sloping line). The slope of a velocity-time graph reveals information about the object's acceleration. If a line crosses the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.

[pic]

The shape of a velocity vs. time graph reveals pertinent information about an object's acceleration. For example, if the acceleration is zero, then the velocity-time graph is a horizontal line (i.e., the slope is zero). If the acceleration is positive, then the line is an upward sloping line (i.e., the slope is positive). If the acceleration is negative, then the velocity-time graph is a downward sloping line (i.e., the slope is negative). If the acceleration is large, then the line slopes up steeply (i.e., the slope is large). Thus, the shape of the line on the graph (horizontal, sloped, steeply sloped, mildly sloped, etc.) is descriptive of the object's motion.

Example 1

Consider a car moving with a constant velocity of +10 m/s. A car which is moving with a constant velocity has an acceleration of 0 m/s/s.

[pic]

The velocity-time data and graph would look like the table and graph below. Note that the line on the graph is horizontal. That is, the slope of the line is 0 m/s/s. Here, it is obvious that the slope of the line (0 m/s/s) is the same as the acceleration (0 m/s/s) of the car.

|Time (s) |[pic] |

|Velocity (m/s) | |

| | |

|0 | |

|10 | |

| | |

|1 | |

|10 | |

| | |

|2 | |

|10 | |

| | |

|3 | |

|10 | |

| | |

|4 | |

|10 | |

| | |

|5 | |

|10 | |

| | |

So in this case, the slope of the line is equal to the acceleration of the velocity-time graph.

Example 2

Consider a car moving with a changing velocity. A car which moves with a changing velocity has an acceleration.

[pic]

The velocity-time data for this motion shows that the car has an acceleration of +10 m/s/s. A graph of this velocity-time data would look like the graph below. Note that the line on the graph is diagonal — that is, it has a slope. The slope of this line, when calculated, is 10 m/s/s. Once again, the slope of the line (10 m/s/s) is the same as the acceleration (10 m/s/s) of the car.

|Time (s) |[pic] |

|Velocity (m/s) | |

| | |

|0 | |

|0 | |

| | |

|1 | |

|10 | |

| | |

|2 | |

|20 | |

| | |

|3 | |

|30 | |

| | |

|4 | |

|40 | |

| | |

|5 | |

|50 | |

| | |

Example 3

Let's examine a more complex case. Consider the motion of a car which travels with a constant velocity (a = 0 m/s/s) of 2 m/s for four seconds and then accelerates at a rate of +2 m/s/s for four seconds. That is, in the first four seconds, the car does not change its velocity (the velocity remains at 2 m/s) then the car increases its velocity by 2 m/s each second over the next four seconds. The velocity-time data and graph are displayed below. Observe the relationship between the slope of the line and the corresponding acceleration value during each four-second interval.

|Time (s) |[pic] |

|Velocity (m/s) | |

| | |

|0 | |

|2 | |

| | |

|1 | |

|2 | |

| | |

|2 | |

|2 | |

| | |

|3 | |

|2 | |

| | |

|4 | |

|2 | |

| | |

|5 | |

|4 | |

| | |

|6 | |

|6 | |

| | |

|7 | |

|8 | |

| | |

|8 | |

|10 | |

| | |

| |

|From 0 s to 4 s: slope = 0 m/s/s |

|From 4 s to 8 s: slope = 2 m/s/s |

A motion such as the one above further illustrates the importance of the principle of slope: the slope of the line on a velocity-time graph is equal to the acceleration of the object. This principle can be used for all velocity-time graphs in order to determine the numerical value of the acceleration.

Check Your Understanding

The velocity-time graph for a two-stage rocket is shown below. Use the graph and your understanding of slope calculations to determine the acceleration of the rocket during the listed time intervals.

a. t = 0 - 1 second

b. t = 1 - 4 seconds

c. t = 4 - 12 seconds

Relating the Shape to the Motion

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Determining the Slope on a v-t Graph

The slope of the line on a velocity vs. time graph is equal to the acceleration of the object.

[pic]

The slope equation says that the slope of a line is found by dividing the amount of rise of the line between any two points by the amount of run of the line between the same two points. In other words:

1. Pick two points on the line and determine their coordinates.

2. Determine the difference in y-coordinates of these two points (rise).

3. Determine the difference in x-coordinates of these two points (run).

4. Divide the difference in y-coordinates (rise) by the difference in x-coordinates (run).

5. Slope = rise/run.

Consider the velocity-time graph . Determine the acceleration (i.e., slope) of the object as portrayed by the graph.

Determining the Area on a v-t Graph

A plot of velocity vs. time can be used to determine the acceleration of an object (slope = acceleration). It can also be used to determine the distance traveled by an object. For velocity vs. time graphs, the area bounded by the line and the axes represents the distance traveled.

The diagram below shows three different velocity-time graphs; the shaded regions between the line and the axes represent the distance traveled during the stated time interval.

The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 6 seconds. This representation of the distance traveled takes on the shape of a rectangle whose area can be calculated using the appropriate equation.

The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 4 seconds. This representation of the distance traveled takes on the shape of a triangle whose area can be calculated using the appropriate equation.

The shaded area is representative of the distance traveled by the object during the time interval from 2 seconds to 5 seconds. This representation of the distance traveled takes on the shape of a trapezoid whose area can be calculated using the appropriate equation.

The method used to find the area under a line on a velocity-time graph depends on whether the section bounded by the line and the axes is a rectangle, a triangle or a trapezoid. Area formulae for each shape are given below.

[pic]

Work out the distance covered for each of the following cases.

[pic]

Calculating the Area of a Triangle

The shaded triangle on the velocity-time graph, has a base of 4 seconds and a height of 40 m/s.

Area of triangle:

A = 0.5 * b * h = (0.5) * (4 s) * (40 m/s) = 80 m.

The object was displaced 80 meters during the first four seconds of motion.

Work out the distance covered for each of the following cases.

[pic]

Calculating the Area of a Trapezoid

The shaded trapezoid on the velocity-time graph, below, has a base of 2 seconds and heights of 10 m/s (on the left side) and 30 m/s (on the right side).

Area of trapezoid:

   A = (0.5) * (b) * (h1 + h2)

      = (0.5) * (2 s) * (10 m/s + 30 m/s) = 40 m.

The object was displaced 40 meters during the time interval from 1 second to 3 seconds.

Now check your understanding by finding the distance traveled by the object in each of the following cases.

[pic]

Alternative Method for Calculating the Area of a Trapezoid

An alternative method of determining the area of a trapezoid involves breaking the trapezoid into a triangle and a rectangle. The areas of the triangle and rectangle are computed individually; the area of the trapezoid is then the sum of the areas of the triangle and the rectangle. This method is illustrated below.

[pic]

Triangle: Area = (0.5) * (2 s) * ( 20 m/s) = 20 m

Rectangle: Area = (2 s) * (10 m/s) = 20 m

Trapezoid: Area = 20 m + 20 m = 40 m

The area bounded by the line and the axes of a velocity-time graph is equal to the displacement of an the object during that time interval. The shaded region can be identified as either a rectangle, triangle, or trapezoid whose area can subsequently be determined.

Graphical Interpretation of Acceleration

Consider a train accelerating from a station along a straight and level track to a maximum speed of 25 m/s in 45 s . It then moves at a constant speed for a further 45 s . It then slowed down to a stop at the next station in 20 s. The easiest way to show this is with a speed time graph.

[pic]

Acceleration is the gradient of the speed-time graph.

From the graph,

• between O and A, the train is accelerating;

• between A and B, the train travels at a constant speed;

• between B and C, the train slows down.  Slowing down can also be called negative acceleration, or deceleration.  It is given a minus sign.

Distance is the area under the speed-time graph.  To work out the total distance, we would add the areas of:

• triangle OAX;

• rectangle ABXY;

• triangle BCY.

What is:

a) The acceleration between O and A

b) The acceleration between B and C

c) The distance covered while the train is at constant speed

d) The total distance.  

e) The average speed?  

The corresponding distance time graph is like this:

[pic]

 We can work out the speed at any instant by measuring the gradient of the distance time graph. The curved line tells us that the speed is changing.

Acceleration is usually uniform, which means that the speed [velocity] is changing at a constant rate. This is shown by a straight line on a speed time graph. However in many real life situations acceleration is not constant. Therefore the graph is not a straight line:

[pic]

Describing Motion with Equations

|Key Words |

|Speed, distance, velocity, displacement, acceleration, time |

 

1. Distance is how far you travel between any two points by any route. It is a scalar quantity.

2. Displacement is the minimum “as the crow flies” distance between two points. It is a vector quantity, so it has direction.

3. Speed is how fast you go, the rate of change of distance.

4. Velocity is rate of change of displacement. It must have a direction.

5. Acceleration can be used as both a vector and a scalar quantity. It is the rate of change of speed or velocity.

|Quantity |Physics Code |Units |

|Distance |s |m |

|Speed at the start |u |m/s |

|Speed at the end |v |m/s |

|Acceleration |a |m/s2 |

|Time |t |s |

 Speed is simply how fast something is going. we measure it in metres per second (written as m/s or ms[pic])

speed (m/s) = distance (m)        

                                    time(s)

Acceleration is the change in velocity per unit time . It is measured by the use of the equation:

                                                [pic]

Where a = acceleration (m/s/s)

           v = final velocity (m/s)

           u = initial (starting) velocity (m/s)

           t = time (seconds)

v - u is the change in velocity

Using the Kinematic Equations

The kinematic equations are a set of four equations which can be utilized to determine unknown information about an object's motion if other details are known. They are also called equations of motion.

1.

Speed at finish = speed at start + change in speed

change in speed = acceleration × time.

Speed at end = speed at start + (acceleration × time)

[pic]

2.

[pic]

Distance = average speed × time

[pic]

4.

[pic]

The application of these four equations to the motion of an object in free fall can be aided by a proper understanding of the conceptual characteristics of free fall motion. These concepts are as follows:

• An object in free fall experiences an acceleration of –9.8 m/s/s. (The negative (–) sign indicates a downward acceleration.)

• If an object is dropped (as opposed to being thrown) from an elevated height to the ground below, the initial velocity of the object is 0 m/s.

• If an object is projected upwards in a vertical direction, it will slow down as it rises upward. The instant at which it reaches the peak of its trajectory, its velocity is 0 m/s.

• If an object is projected upwards in a vertical direction, then the velocity at which it is projected is equal in magnitude and opposite in sign to the velocity it has when it returns to the same height. That is, a ball projected vertically with an upward velocity of +30 m/s will have a downward velocity of –30 m/s when it returns to that same height.

•

Free Fall and the Acceleration of Gravity

Introduction to Free Fall

A free-falling object is an object which is falling under the sole influence of gravity. Thus, any object which is moving and being acted upon only by the force of gravity is said to be "in a state of free fall."

• All free-falling objects (on Earth) accelerate downwards at a rate of approximately 10 m/s/s (to be exact, 9.8 m/s/s).

The Acceleration of Gravity

A free-falling object has an acceleration on Earth of 9.8 m/s/s, downward. It is known as the acceleration of gravity . This quantity is such an important quantity that physicists have a special symbol to denote it – the symbol g.

In Physics we will use the approximated value of 10 m/s/s. g = 10 m/s/s, downward

The acceleration is the rate at which an object changes its velocity. Between any two points in an object's path, acceleration is the ratio of velocity change to the time taken to make that change. To accelerate at 10 m/s/s means to change your velocity by 10 m/s each second.

[pic]

.

Position vs. Time Graphs

The position vs. time graph for a free-falling object is shown below.

[pic]

Observe that the line on the graph is curved. A curved line on a position vs. time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration of g = 10 m/s/s (approximate value), you would expect that its position-time graph would be curved. A closer look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object, the initial small slope indicates a small initial velocity and the final large slope indicates a large final velocity.

Velocity vs. Time Graphs

The velocity vs. time graph for a free-falling object is shown below.

[pic]

Observe that the line on the graph is a straight, diagonal line. A diagonal line on a velocity vs. time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration of g = 10 m/s/s (approximate value), you would expect that its velocity-time graph would be diagonal. A closer look at the velocity-time graph reveals that the object starts with a zero velocity (starts from rest) and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object which is moving in the negative direction and speeding up is said to have a negative acceleration. Since the slope of any velocity vs. time graph is the acceleration of the object, the constant, negative slope indicates a constant, negative acceleration.

Free Fall and the Acceleration of Gravity

How Fast? and How Far?

Free-falling objects are in a state of acceleration. Specifically, they are accelerating at a rate of 10 m/s/s. This is to say that the velocity of a free-falling object is changing by 10 m/s every second. If dropped from a position of rest, the object will be traveling 10 m/s at the end of the first second, 20 m/s at the end of the second second, 30 m/s at the end of the third second, etc.

How Fast?

The velocity of a free-falling object which has been dropped from a position of rest is dependent upon the length of time for which it has fallen. The formula for determining the velocity of a falling object after a time of t seconds is:

v = g * t

where g is the acceleration of gravity.

Example

t = 6 s

v = (10 m/s2) * (6 s) = 60 m/s

t = 8 s

v = (10 m/s2) * (8 s) = 80 m/s

How Far?

The distance which a free-falling object has fallen from a position of rest is also dependent upon the time of fall. The distance fallen after a time of t seconds is given by the formula below:

d = 0.5 * g * t2

where g is the acceleration of gravity (approximately 10 m/s/s on Earth; its exact value is 9.8 m/s/s). The equation above can be used to calculate the distance traveled by the object after a given amount of time.

Example

t = 1 s

d = (0.5) * (10 m/s2) * (1 s)2 = 5 m

t = 2 s

d = (0.5) * (10 m/s2) * (2 s)2 = 20 m

t = 5 s

d = (0.5) * (10 m/s2) * (5 s)2 = 125 m

The diagram below (not drawn to scale) shows the results of several distance calculations for a free-falling object dropped from a position of rest.

 2. A brick falls off the top of a wall under construction and drops into a bed of sand 14.5 m below. It makes a dent in the sand 185 mm deep. What is:

a)       The speed of the brick just before it hits the sand.

b)       Its deceleration in the sand.

c)       What would happen to a person undergoing that deceleration?

Thinking, Braking & Total stopping distance.

Road users are advised to maintain safe distances to cut down the risk of accidents. The shortest stopping distance of a vehicle depends on its speed and on the road conditions.

Stopping is made up of two parts: thinking and braking.

Thinking time is the reaction time, when your brain is responding to the hazard ahead of you. Thinking distance is the distance travelled by the car in the time it takes the driver to react.

Factors affecting thinking time.

1. Tiredness: Your brain thinks slower - you will not be able to apply the brakes as quickly.

2. Alcohol : Being under the influence - even legally - seriously alters how well you can judge hazards. Your body also moves less accurately. Late or missed braking results!

3. Drugs : Most drugs make you less alert and less aware of hazards. Even legal pain-killers and hay-fever tablets can seriously affect reaction times.

4. Distractions : In-car distractions (e.g. very loud music, mobile phones, crying babies, etc.) take your mind off the road ahead.

Braking time is the time taken to slow the vehicle down from your initial speed to zero . The Braking distance is the distance traveled by the car from the point where the brakes are applied to where it comes to rest.

These are some of the factors that affect how effective your braking will be:

• Brakes : Damaged brakes won't work as well, so you'll need to brake for longer.

• Tyres : Good tyres can reduce braking distance by many metres! Worn tyres (with little tread) will have good grip in the dry but in the wet will lead to much longer braking distances.

• Road Surface : Different types of surface provide different levels of grip, especially in the wet. If the road is wet, braking distance will always be longer. Oil spills on the road, gravel, etc. all reduce grip and increase braking distances.

Stopping time is the thinking and braking times added together. The total time to stop a moving vehicle. Stopping distance is the thinking distance added to the braking distance.

The following graph illustrates how each of these distances increases with speed.

1. The thinking distance is proportional to the speed. This is because the vehicle travels at constant speed during the reaction period before the brakes are applied.

2. The braking distance is proportional to the square of the speed.

[pic]

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