introduction to motion homework

20 Introduction to Motion

Objects are in motion everywhere we look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. And even in inanimate objects, there is continuous motion in the vibrations of atoms and molecules. Questions about motion are interesting in and of themselves: How long will it take for a space probe to get to Mars? Where will a football land if it is thrown at a certain angle? But an understanding of motion is also key to understanding other concepts in physics. An understanding of acceleration, for example, is crucial to the study of force.

Our formal study of physics begins with kinematics which is defined as the study of motion without considering its causes . The word “kinematics” comes from a Greek term meaning motion and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). For now, we will study only the motion of a football, for example, without worrying about what forces cause or change its motion. Such considerations come in other chapters. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one-dimensional motion. Later, we apply concepts developed here to study motion along curved paths (two- and three-dimensional motion); for example, that of a car rounding a curve.

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Problems & Exercises

1.2 physical quantities and units.

The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this?

A car is traveling at a speed of 33 m/s 33 m/s size 12{"33"" m/s"} {} . (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h 90 km/h size 12{"90"" km/h"} {} speed limit?

Show that 1 . 0 m/s = 3 . 6 km/h 1 . 0 m/s = 3 . 6 km/h size 12{1 "." 0`"m/s"=3 "." "6 km/h"} {} . Hint: Show the explicit steps involved in converting 1 . 0 m/s = 3 . 6 km/h. 1 . 0 m/s = 3 . 6 km/h. size 12{1 "." 0`"m/s"=3 "." "6 km/h"} {}

American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)

Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.)

What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.)

Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.)

The speed of sound is measured to be 342 m/s 342 m/s size 12{"342"" m/s"} {} on a certain day. What is this in km/h?

Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometers per million years?

(a) Refer to Table 1.3 to determine the average distance between the Earth and the Sun. Then calculate the average speed of the Earth in its orbit in kilometers per second. (b) What is this in meters per second?

1.3 Accuracy, Precision, and Significant Figures

Express your answers to problems in this section to the correct number of significant figures and proper units.

Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?

A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty?

(a) A car speedometer has a 5.0 % 5.0 % size 12{5.0%} {} uncertainty. What is the range of possible speeds when it reads 90 km/h 90 km/h size 12{"90"" km/h"} {} ? (b) Convert this range to miles per hour. 1 km = 0.6214 mi 1 km = 0.6214 mi size 12{"1 km" "=" "0.6214 mi"} {}

An infant’s pulse rate is measured to be 130 ± 5 130 ± 5 size 12{"130" +- 5} {} beats/min. What is the percent uncertainty in this measurement?

(a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats does he or she have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y?

A can contains 375 mL of soda. How much is left after 308 mL is removed?

State how many significant figures are proper in the results of the following calculations: (a) 106 . 7 98 . 2 / 46 . 210 1 . 01 106 . 7 98 . 2 / 46 . 210 1 . 01 size 12{ left ("106" "." 7 right ) left ("98" "." 2 right )/ left ("46" "." "210" right ) left (1 "." "01" right )} {} (b) 18 . 7 2 18 . 7 2 size 12{ left ("18" "." 7 right ) rSup { size 8{2} } } {} (c) 1 . 60 × 10 − 19 3712 1 . 60 × 10 − 19 3712 size 12{ left (1 "." "60" times "10" rSup { size 8{ - "19"} } right ) left ("3712" right )} {} .

(a) How many significant figures are in the numbers 99 and 100? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties?

(a) If your speedometer has an uncertainty of 2 . 0 km/h 2 . 0 km/h size 12{2 "." 0" km/h"} {} at a speed of 90 km/h 90 km/h size 12{"90"" km/h"} {} , what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/h 60 km/h size 12{"60"" km/h"} {} , what is the range of speeds you could be going?

(a) A person’s blood pressure is measured to be 120 ± 2 mm Hg 120 ± 2 mm Hg size 12{"120" +- 2" mm Hg"} {} . What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg 80 mm Hg size 12{"80"" mm Hg"} {} ?

A person measures his or her heart rate by counting the number of beats in 30 s 30 s size 12{"30"" s"} {} . If 40 ± 1 40 ± 1 size 12{"40" +- 1} {} beats are counted in 30 . 0 ± 0 . 5 s 30 . 0 ± 0 . 5 s size 12{"30" "." 0 +- 0 "." 5" s"} {} , what is the heart rate and its uncertainty in beats per minute?

What is the area of a circle 3 . 102 cm 3 . 102 cm size 12{3 "." "102"" cm"} {} in diameter?

If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22-mi marathon?

A marathon runner completes a 42 . 188 -km 42 . 188 -km size 12{"42" "." "188""-km"} {} course in 2 h 2 h size 12{2" h"} {} , 30 min, and 12 s 12 s size 12{"12"" s"} {} . There is an uncertainty of 25 m 25 m size 12{"25"" m"} {} in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?

The sides of a small rectangular box are measured to be 1 . 80 ± 0 . 01 cm 1 . 80 ± 0 . 01 cm size 12{1 "." "80" +- 0 "." "01"" cm"} {} , {} 2 . 05 ± 0 . 02 cm, and 3 . 1 ± 0 . 1 cm 2 . 05 ± 0 . 02 cm, and 3 . 1 ± 0 . 1 cm size 12{2 "." "05" +- 0 "." "02"" cm, and 3" "." 1 +- 0 "." "1 cm"} {} long. Calculate its volume and uncertainty in cubic centimeters.

When non-metric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was employed, where 1 lbm = 0 . 4539 kg 1 lbm = 0 . 4539 kg size 12{1" lbm"=0 "." "4539"`"kg"} {} . (a) If there is an uncertainty of 0 . 0001 kg 0 . 0001 kg size 12{0 "." "0001"`"kg"} {} in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?

The length and width of a rectangular room are measured to be 3 . 955 ± 0 . 005 m 3 . 955 ± 0 . 005 m size 12{3 "." "955" +- 0 "." "005"" m"} {} and 3 . 050 ± 0 . 005 m 3 . 050 ± 0 . 005 m size 12{3 "." "050" +- 0 "." "005"" m"} {} . Calculate the area of the room and its uncertainty in square meters.

A car engine moves a piston with a circular cross section of 7 . 500 ± 0 . 002 cm 7 . 500 ± 0 . 002 cm size 12{7 "." "500" +- 0 "." "002"`"cm"} {} diameter a distance of 3 . 250 ± 0 . 001 cm 3 . 250 ± 0 . 001 cm size 12{3 "." "250" +- 0 "." "001"`"cm"} {} to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.

1.4 Approximation

How many heartbeats are there in a lifetime?

A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?

How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10 − 22  s 10 − 22  s size 12{"10" rSup { size 8{ - "22"} } " s"} {} .)

Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of 10 − 27  kg 10 − 27  kg size 12{"10" rSup { size 8{ - "27"} } " kg"} {} and the mass of a bacterium is on the order of 10 − 15  kg. 10 − 15  kg. size 12{"10" rSup { size 8{ - "15"} } "kg"} {} )

Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom?

(a) What fraction of Earth’s diameter is the greatest ocean depth? (b) The greatest mountain height?

(a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?

Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

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1.1: Introduction to One-Dimensional Kinematics

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Objects are in motion everywhere we look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. And even in inanimate objects, there is continuous motion in the vibrations of atoms and molecules. Questions about motion are interesting in and of themselves:  How long will it take for a space probe to get to Mars?  But an understanding of motion is also key to understanding other concepts in physics. An understanding of acceleration, for example, is crucial to the study of force.

Our formal study of physics begins with  kinematics  which is defined as the  study of motion without considering its causes . The word “kinematics” comes from a Greek term meaning motion and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). In one-dimensional kinematics we will study only the  motion  of a football, for example, without worrying about what forces cause or change its motion. Such considerations come in other chapters. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one-dimensional motion.

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Unit 1: One-dimensional motion

About this unit, physics foundations.

• Introduction to physics (Opens a modal)
• Preparing to study physics (Opens a modal)
• Intro to vectors and scalars (Opens a modal)

Distance, displacement, and coordinate systems

• Distance and displacement introduction (Opens a modal)
• Distance and displacement in one dimension (Opens a modal)
• Position-time graphs (Opens a modal)
• Worked example: distance and displacement from position-time graphs (Opens a modal)
• Distance and displacement review (Opens a modal)
• Finding distance and displacement from graphs Get 3 of 4 questions to level up!

Average velocity and average speed

• Average velocity and speed worked example (Opens a modal)
• Average velocity and speed review (Opens a modal)
• Average velocity and speed in one direction: word problems Get 3 of 4 questions to level up!
• Average velocity and speed with direction changes: word problems Get 3 of 4 questions to level up!

Velocity and speed from graphs

• Instantaneous speed and velocity (Opens a modal)
• Why distance is area under velocity-time line (Opens a modal)
• Instantaneous velocity and speed from graphs review (Opens a modal)
• Average velocity and average speed from graphs Get 3 of 4 questions to level up!
• Instantaneous velocity and instantaneous speed from graphs Get 3 of 4 questions to level up!
• Finding displacement from velocity graphs Get 3 of 4 questions to level up!

Average and instantaneous acceleration

• Acceleration (Opens a modal)
• What are velocity vs. time graphs? (Opens a modal)
• Acceleration review (Opens a modal)
• Calculating average acceleration from graphs Get 3 of 4 questions to level up!
• Connecting acceleration and velocity graphs Get 3 of 4 questions to level up!

Motion with constant acceleration

• Choosing kinematic equations (Opens a modal)
• Airbus A380 take-off distance (Opens a modal)
• Motion with constant acceleration review (Opens a modal)
• Choosing the best kinematic equation Get 3 of 4 questions to level up!
• Kinematic equations: numerical calculations Get 3 of 4 questions to level up!

Objects in freefall

• Plotting projectile displacement, acceleration, and velocity (Opens a modal)
• Impact velocity from given height (Opens a modal)
• Freefall review (Opens a modal)
• Freefall: graphs and conceptual questions Get 3 of 4 questions to level up!
• Solving freefall problems using kinematic formulas Get 3 of 4 questions to level up!

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Introduction to Motion

The study of motion is often called kinematics.  We will begin our study with one dimensional kinematics.   We will later expand to 2 and 3 dimensional kinematics after we have studied vectors.

We can give the position of an object in relation to a reference point.  There are a number of variables we can use for position, such as x, d , or s .   The official metric unit for position is the meter ( abbreviated m) .  The meter was first defined in terms of the circumference of the Earth on a meridian passing through Paris.  It is now defined in terms of the speed of light.   When working with other scales, it might be convenient to use other metric units such as the nanometer (nm) , the centimeter (cm) , and the kilometer (km) .

Exponential Notation

We will often use exponential notation . Exponential notation is convenient for expressing very large and small numbers.

For instance, 12,300 would be expressed as  1.23 x 10,000 or 1.23 x 10 4

So 3.14 km = 3140 m = 3.14 x 10 3 m

For small numbers, 0.000345 = 3.45 x 10 -4

A micrometer, 1 μm = 10 -6 m The width of a human hair on average is 10 μm .  This would be 10 x 10 -6 m .

The wavelength of a helium-neon laser is 633 nm = 633 X 10 -9 m = 6.33 x 10 -7 m

Metric Units

The common metric units are given in powers or 3.

The kilometer is 1000 m .

Although the 100 centimeters = 1 meter it is not actually a common unit.

1 Millimeter = 1mm = 10- 3 m

1 Micrometer = 1um = 10 -6 m

1 Nanometer = 1nm = 10 -9 m

1 Picometer = 1pm = 10 -12 m

1 Femotometer = 1fm = 10 -15 m also known as a Fermi

Except for kilometer, we often do not use the larger metric prefixes for distance.  But they are used for frequencies and other units in physics.

1 Kilometer = 1 km = 1000 m = 10 3 m

Megameter = 1Mm = 10 6 m

Gigameter = 1Gm = 10 9 m

Terrameter = 1 Tm = 10 12 m

Imperial Units

Common British Imperial units for measuring distance include the inch, the foot, the yard, and the mile. An easy way to remember the conversion from meters to miles can be remembered in terms of Track and Field.  The loop in a track is ¼ mile long.  It is also known as the 400 m race, so 1 mile is approximately = 1600 m .   Engineers in America commonly use Imperial units. Very small measurements for the purposes of manufacturing are given in 1/1000ths of an inch.

When dealing with astronomical distances there are other units we might use such as the light-year, the parsec, or the Astronomical Unit. The light-year is the distance light will travel in one year.  An object which is one parsec away has one arc-second of parallax  from Earth.  An astronomical unit is the average distance from the Earth to the Sun.

Distance vs Displacement

In physics we often study the change in position of an object.  If we are only examining the change in position from the start of our observation to the end, we are talking about displacement .  We ignore how we get from point A to point B.  We are only concerned with how the crow flies.  If we are concerned with our path, we are working with distance (see figure A).

For example, let us suppose I were to talk around the perimeter of a square classroom (see figure B).  The classroom is 10 meters on a side.  At the end of my trip I return to my original starting position.  The distance traveled would be 40 m.  The displacement would be zero meters because displacement only depends on the starting and ending positions.

The other important distinction between distance and displacement is that distances do not have a direction.  If you were wearing a pedometer is would record distance.  The odometer on a car records distance.  Displacement has a direction and a magnitude .  Magnitude is a fancy physics term for size or amount.

For instance, suppose I walked 10 m North, 10 m East, 10 South, and then 5 m West (see figure C).  My distance traveled would be 35 m. There is a magnitude but no net direction.  Since we can describe distance with just a magnitude (but no direction) we call it a scalar .    But my displacement would be 5 m due East.  As displacement has both a magnitude and a direction, we call it a vector .

We measure time in seconds .  We will use the variable t for time.    The elapsed time for a certain action would be  ΔT.   The Greek letter delta, Δ,  is used to represent a change in a quantity.  If we are talking about a reoccurring event (such as the orbit of the Earth around the sun) we talk about the Period of time T , with a capitol T.

For longer periods of time we will often use the conventional minutes, hours, days, or years.  For shorter periods of time will often use exponential notation or may use milliseconds, microseconds, picoseconds, or femtoseconds.

Speed and Velocity

Building on changes in position and changes in time, we can examine the rate at which these changes in position take place. How fast are we moving?  You probably use the terms speed and velocity interchangeably in your everyday vernacular, but in physics they have distinct meanings.    Speed is a scalar and has no direction.  Speed can be  defined as

speed = distance/elapsed time

Velocity is a vector.  We could consider velocity to be speed in a given direction.  To calculate the average velocity over a period of time, we use displacement and elapsed time.

Where v is velocity, x is position, t is time.  The Greek letter delta, Δ, means a change in a quantity, such as the change in position or the change in time.  The bar over the velocity v means the terms in averaged. For instance,  Δx = x f – x o , or the change is position equals the difference of the final position and the original positions.

Our first set of problems will involve the above kinematic equation.

Problem Solving Method

When solving physics problems, it is useful to follow a simple  problem solving strategy.   Although at first, it may be easy to solve some problems in your head, by following this strategy you will develop good problem solving habits.  Just as you must develop good habits by brushing your teeth every day, you should attempt to follow the following methodology for solving physics problems.  The first step is Step 0 because it does not always apply.

Step 0:  Draw a picture of the problem if appropriate.

Step 1: Write down the given information

Step 2: Write down the unknown quantity you are trying to find out

Step 3: Write down the physics equations or relationships that will connect your given information to the unknown variables.

Step 4:  Perform algebraic calculations necessary to isolate the unknown variable.

Step 5: Plug in the given information to the new equation.  Cancel appropriate units and do the arithmetic.

Example problems:

Example 1: a robot travels across a countertop a distance of 88.0 cm, in 30 seconds.  what is the speed of the robot.

In this case, we do not need to do any algebra.

Significant figures:  At this point we should not how many significant figures our answer has.  Your final answer cannot have more information that your original data.  We were presented with a distance and a time with only 3 significant figures, therefore our final answer cannot have more precision than this.

Now let us look at a problem which does require some algebra.

Example 2: The SR-71 Blackbird could fly at a speed of Mach 3,  or 1,020 m/s.  How much time would it take the SR-71 to take off from Los Angeles and fly to New York City via a path which is a distance of 5500 miles.

You should note that you need to convert miles to meters, remembering that 1 mile = 1600 m

First we need to algebraically isolate the variable t .First we multiply both sides by t, and t cancels on the right hand side of the equation.

Then dividing both sides by v gives us

Now, plugging in for distance and speed gives us

Note the units and the number of significant digits.  Because one piece of our original data (distance) only had two significant digits, we have to round off our final answer to 2 significant digits.  Also, look at the cancelation of units.  The meters in the units cancel.  Our units have the reciprocal of a reciprocal, thus the final units are in seconds, which you might have guessed since we are working with time.  For ease of perspective we converted these units into minutes.

Average velocity vs instantaneous velocity

Another important distinction is finding an average value or the velocity versus the velocity at a given instant in time.  To find an average velocity we only the measure the change in position and the total elapsed time.

However, finding the velocity at a given instant can be tricky.  The elapsed time for an instant has no finite length.  Similarly, a physical position in space has no finite size.  To calculate this using equations we would have to reduce the elapsed time to a near infinitesimally small amount of time.  Mathematically, this is the basic for calculus which was developed separately by both Newton  and Leibniz.  In standard calculus notation we would say the instantaneous velocity can be expressed as

In our next lesson we will learn how to determine the instantaneous velocity using graphical techniques.

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