MOTION IN ONE DIMENSION

Distance and Displacement

Speed and Velocity

Acceleration

Motion with Constant Acceleration

Free Fall and the Acceleration Due to Gravity

Review Problems for Chapter 1

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The study of the motion of an object is called __mechanics__,
and we begin this course by examining a part of mechanics called __kinematics__.
Kinematics is the study of motion exclusive of the influence of mass and force.
In this chapter we begin by introducing and defining the fundamental variables
of kinematics. We will then examine motion in one dimension with constant acceleration.
The equations developed to describe this motion will allow us to predict the
motion of objects with constant acceleration. The chapter ends with a study
of free fall and the constant acceleration due to gravity. This is the most
common example of constant acceleration.

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Distance and displacement are both used to describe the extent
of a body’s motion. __Distance__ is simply the length of the path a body
follows in moving from one point to another. __Displacement__ is the straight
line distance between the initial and final positions of the body.

These concepts are best understood by looking at some simple examples. Marquette, Michigan lies 300 miles north-northwest of Lansing, the state capital. If you were to drive from Lansing to Marquette you must first drive 250 miles north and then 150 miles west. This is because Lake Michigan lies between the two cities. As a result, the distance you would travel would be 400 miles. However, your displacement would only be 300 miles.

As a second example consider the outdoor mile run. On a
quarter mile track you would need to complete four laps to run a distance of
one mile. However your displacement would be zero since you end at the same
place you start.

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Speed is the most basic property of a moving body. Be virtue of its motion a body travels a certain distance in a given time. An automobile, for example, travels so many miles per hour.

Example:

What distance would you cover if you traveled at an average speed of 50 mph for 3 hours?

This is a rather simple problem and with little trouble you realize at once that the answer is 150 miles.

To solve the problem above you have just used your first equation:

Distance = ( average speed ) x ( time )

Or in symbols

(1.1)

where the bar above the u indicates that it is an average value.

With simple algebra Equation 1.1 can be written in a form which defines average speed.

(1.2)

Whenever a quantity is divided by the time we speak about the “rate: at which this quantity changes.

So speed is a measure of how quickly or slowly distance is covered.

Speed is the rate at which distance is covered.

__Note__: The units of
speed are always a length divided by a time unit (mi/hour, m/sec, ft/sec).

Examples:

1. If the Lansing to Marquette trip (d = 400 miles) is completed in eight hours what is the average speed?

2. The speed of sound is about 1100 ft/sec. If the time between a flash of lightning and its thunderclap is 5 seconds, how far away is the lightning?

In the first example we found the average speed during the trip from Lansing
to Marquette to be 50 mph. However, at some times the car was probably traveling
faster, slower, of completely stopped. If we wanted to know the speed of the
car at any particular time we would simply read the speedometer. This tells
us our speed at that instant. We define the speed an object has at a certain
instant to be the object’s __instantaneous speed__.

u = instantaneous speed

__Note__:
No bar above the u indicates an instantaneous value.

You have probably used the words speed and velocity interchangeably. However, velocity is not the same as speed. Velocity specifies the direction a body is moving, as well as how fast it moves.

n = instantaneous velocity

Quantities, such as velocity,
which specify direction and magnitude are called __vectors__.

Questions:

1. Consider the Lansing to Marquette trip. If the speedometer read 55 mph going through Mt. Pleasant what was the instantaneous velocity?

2. What was the average velocity for the entire trip?

3. If a body travels at constant
speed does it follow that it is traveling at constant velocity? What about
the reverse?

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Up to this point we have considered cases where the velocity is constant, or, if changes, we looked only at the average velocity. An object whose velocity is changing is said to be accelerating.

Define acceleration at the rate at which velocity changes.

Or in symbols,

(1.3)

Where n _{F} = final
velocity and n _{I}
= initial velocity.

__Example__: A runner
starting with a velocity of 4 ft/sec accelerates to a velocity if 10 ft/sec
in 3 seconds. What is the acceleration?

Note: This answer is read “2 feet per second each second” or “2 feet per second
per second.” In other words the speed of the runner is increasing at a rate
of 2 ft/sec each second, and this constant rate of increase in velocity is the
quantity that we have called acceleration. The above result is usually written
as 2 ft/sec^{2} and is read “2 feet per second
squared.”

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Relations between position, velocity, and acceleration can always be worked out, but this can be very difficult if the motion is the least bit complicated. One very important case that is not complicates is motion with constant acceleration.

Let us first recall the equations that defined average velocity (1.2) and acceleration (1.3).

In the special case of constant acceleration the average acceleration is also the instantaneous acceleration, and we can write

Simple algebra gives

(1.4)

Furthermore, when the acceleration is constant, the speed and velocity of the moving body change uniformly with time. Because of this, the average velocity is merely one-half the sum of if the initial and final velocities; thus

(1.5)

Now if we substitute Eq (1.2) in Eq (1.5), we find

And rearranging gives

(1.6)

We may use Eq (1.4) along with Eq (1.6) to eliminate the final velocity.

(1.6)

Finally, we will use Eq (1.3) and Eq (1.6) to eliminate time. Starting with Eq (1.6) we have

And rearranging gives

We now substutute this into Eq (1.3)

Finally, (1.8)

We now have four equations available to use in the solution of problems involving uniformly accelerated motion. They are

__Note__: In solving the problems of motion there
are five variables: position, time, acceleration, and initial and final velocities.
Each of the four equations above contain exactly four of these variables. It
is necessary, therefore, to know at least three of the variables.

Examples:

1. A car starting from rest attains a speed of 28 m/sec in 20 sec. Find the acceleration of the car and the distance it travels in this time.

2.
A bicyclist accelerates at a rate of 4 ft/sec^{2}. If
she starts at rest and accelerates at this rate for 5 seconds what is her final
velocity and what distance does she go?

3. How long does it take a car going
30 m/sec to stop of it decelerates at 7 m/sec^{2}?

The most common example of constant acceleration is free fall. In the 17^{th}
century Galileo showed that if one neglects air-friction effects all bodies
fall to earth in the same way, independent of their weight. Although the exact
value of this acceleration resulting from the pull of gravity varies slightly
from place to place on the earth, it is close to 9.8 m/sec^{2},
which is the same as 32.2 ft/sec^{2}.

In this class we will use the approximate values of 10m/sec^{2}
and 32 ft/sec^{2} for the acceleration of gravity.
This will greatly ease our calculations, and the error introduced by this approximation
is small.

__C A U T I O N !__

For many students the biggest problems when using the kinematics equations are the signs of the quantities. It is clear that when an object accelerates its acceleration should be positive (its velocity and acceleration are in the same direction), and when an object decelerates its acceleration should be negative (its velocity and acceleration are in opposite directions). But what should one do when the problems involve both upward and downward motion? To resolve this problem, keep in mind that velocity and acceleration are vector quantities. The sign in front of these quantities specifies their direction, and you have the choice of defining what direction is positive or negative. However, remember that the acceleration of gravity is always down and your choice of signs will never change this fact.

Examples:

1. A ball is thrown straight up with an initial velocity of 30 m/sec. How high does it rise?

Since the motion of the ball is always up, the logical choice is to call all upward quantities positive.

2. A ball is dropped from a tall building and strikes the ground 4 seconds later. With what velocity does it strike the ground and what distance does it fall?

Since the motion of the ball is always down it makes sense to call all downward quantities positive.

3. Repeat the previous problem. This time see what happens if up is used as the positive direction.

In example 2, down was chosen to be the positive direction and the results
were that the ball traveled down 80 m and struck the ground traveling down at
a velocity of 40 m/sec. In example 3, we let up be the positive direction and
the results were d = -80 m and v_{F} = -40 m/sec.
These negative signs tell us the ball traveled down. In both examples the results
are in fact the same.

4. A ball is thrown down with an initial velocity of 20 ft/sec from a height of 50 ft. Find the velocity it has when it strikes the ground.

Since the motion is always down, pick down to be the positive direction. Also, be careful. The units are English.

Distance Average and Instantaneous Quantities

Displacement Motion with Constant Acceleration

Speed Free Fall

Velocity The Acceleration of Gravity

Acceleration

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1. What is the length if time for a 90 mile/hour fastball to cover the approximately 60 feet from the pitcher’s hands to the plate? How about a curve ball at 60 mile/ hour? Express your answers in seconds. (Hint: 1 mph = 1.47 ft/sec).

2. Estimate the difference in times for a left-handed batter and a right-handed batter to reach first base. Assume the average velocity of the runner is 20 feet/second and the extra distance a right-handed batter must run is 4 ft.

3. Estimate you average running speed in ft/sec and mile/hr. Do this for both sprinting and long distance running.

4. In the 1932 Olympic Games, the 3,000 meter steeplechase was won by Volmari Iso-Hollo of Finland. He ran 3450 meters in 10 min, 33.4 seconds. (They ran an extra lap because the official made a counting error.) What was his average speed in this race?

5. In the 1984 Winter Olympics Bill Johnson won the men’s downhill with a time of 105.59 seconds. The race was 10,032 ft. (1.9 miles) long. What was his average speed?

Peter Mueller of Switzerland placed second in this event with a time of 105.86 seconds. What was Peter Mueller’s average speed?

By comparing these numbers what can you say about the competiveness of world class skiers? Does it really make a difference if they go to the trouble of streamlining their equipment and themselves?

6. A sprinter in a 100 meter race passes the 50 meter mark with a constant velocity of 10 m/sec.

- Where would he be 2.0 seconds later?
- Would you expect his time for the race to be more than 10 sec? Why?

7. A cyclist from rest is able to reach a speed of 30 ft/sec in 10 seconds. What is the acceleration of the cyclist and what distance does he cover in this time?

8. The average driver has a reaction time of 0.8 sec. On dry pavement a car
can decelerate at a rate of 20 ft/sec^{2}.

- Guess what distance is needed to bring a 60 mph (88 ft/sec) car to rest.
- Calculate the distance a car travels in a time of 0.8 sec if it is moving at a constant velocity of 60 mph (88 ft/sec).
- Calculate the distance a car travels if its initial velocity is 60 mph and
it decelerates at a rate of 20 ft/sec
^{2}. - Add the results in parts b and c and compare to your guess in part a.
- Repeat if the initial velocity is 80 mph (117 ft/sec).

9. A ball is thrown straight up with a velocity of 30 m/sec (67 mph).

- How high does it rise?
- How long does it take to reach its maximum height?
- How long does it take to rise and return?
- What is it velocity when it returns?

10. A student stands on the edge of a building and throws a ball upward with a velocity of 20 m/sec. What is the location and velocity of the ball when t = 1, 2, 3, 4, and 5 seconds?

11. World class pole vaulters can attain heights of almost 20 ft. Assuming that they fall 16 ft to the padding below calculate the time it takes them to fall and their velocity when they strike the pad.

12. a. An outstanding jumper can jump vertically about 42 inches (3.5 ft). How long would this individual be in the air?

b. Most people can run and jump only 2 ft vertically. How long would
such an individual be in the air?

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