Kinematics

MECHANICS

Our study of physics begins in the area of mechanics, which deals with the relations among force, matter, and motion. Mechanics consists of: kinematics, dynamics and static.

Kinematics is the study of motion of objects without taking into account the causes of their motion.

Dynamics is the study of motion of objects by taking into account the cause of their change of state (rest or of uniform motion).

Static is study of the object at rest under the effect of forces in equilibrium.

Motion is a continuous change of the position of a body with respect to its surroundings

In physics we are concerned with three types of motion: translational, rotational, and vibrational. A car moving down a highway is an example of translational motion, the Earth's spin on its axis is an example of rotational motion, and the back-and-forth movement of a pendulum is an example of vibrational motion.

In our study of translational motion, we describe the moving object as a particle regardless of its size. In general a particle is a point-like mass having infinitesimal size. For example, if we wish to describe the motion of the Earth around the Sun, we can treat the Earth as a particle and obtain reasonably accurate data about its orbit. Wesay that the particle is a model for a moving body; it provides a simplified, idealized description of the position and motion of the body.

1-1 Position and Displacement

To locate an object means to find its position relative to some reference point, often the origin(or zero point) of an axis such as the axis in Fig. 2-1. The positive directionof the axis is in the direction of increasing numbers (coordinates), which is toward the right in Fig. 1. The opposite direction is the negative direction.

Fig. 1-1

For example, a particle might be located at m, which means that it is 5 m in the positive direction from the origin. If it were at m, it would be just as far from the origin but in the opposite direction.

A change from one position to another position is called a displacement, where

(1)

(The symbol , the Greek uppercase delta, represents a change in a quantity, and it means the final value of that quantity minus the initial value.) When numbers are inserted for the position values and , a displacement in the positive direction (toward the right in Fig. 1) always comes out positive, and one in the opposite direction (left in the figure), negative. For example, if the particle moves from = 5m to = 12 m, then . The positive result indicates that the motion is in the positive direction. If the particle then returns to m, the displacement for the full trip is zero. The actual number of meters covered for the full trip is irrelevant; displacement involves only the original and final positions.

A plus sign for a displacement need not be shown, but a minus sign must always be shown. If we ignore the sign (and thus the direction) of a displacement, we are left with the magnitude(or absolute value) of the displacement. In the previous example, the magnitude of is 7 m.

Displacement is an example of a vector quantity,which is a quantity that has both a direction and a magnitude. Here all we need is the idea that displacement has two features: (1) Its magnitude is the distance (such as the number of meters) between the original and final positions. (2) Its direction, from an original position to a final position, can be represented by a plus sign or a minus sign if the motion is along a single axis.

•CHECKPOINT 1:Here are three pairs of initial and final positions, respectively, along an axis. Which pairs give a negative displacement: (a) ; (b) ; (c) ?

Exercises

1. A watermelon seed has the following coordinates: m, m, and m. Find its position vector (a) in unit-vector notation and as (b) a magnitude and (c) an angle relative to the positive direction of the axis, (d) Sketch the vector on a right-handed coordinate system. If the seed is moved to the xyz coordinates (3.00 m, 0 m, 0 m), what is its displacement (e) in unit-vector notation and as (f) a magnitude and (g) an angle relative to the positive direction of the x axis?

2. The position vector for an electron is . (a) Find the magnitude of . (b) Sketch the vector on a right-handed coordinate system.

3, The position vector for a proton is initially and then later is , all in meters. (a) What is the proton's displacement vector, and (b) to what plane is that vector parallel?

1-2 Average Velocity and Average Speed

A compact way to describe position is with a graph of position plotted as a function of time - a graph of . (The notation represents a function

Fig. 1-2

of , not the product times .)As a simple example, Fig. 1-2 shows the position function for a stationary armadillo (which we treat as a particle) at m.

Fig.2-2. The graph of for an armadillo that is stationary at m. The value of is m for all times

Figure 1-3a, b

Figure 1-3a, also for an armadillo, is more interesting, because it involves motion. The armadillo is apparently first noticed at when it is at the position m. It moves toward , passes through that point at s, and then moves on to increasingly larger positive values of .

Figure 1-3bdepicts the actual straight-line motion of the armadillo and is something like what you would see. The graph in Fig. 1-3ais more abstract and quite unlike what you would see, but it is richer in information. It also reveals how fast the armadillo moves.

Actually, several quantities are associated with the phrase "how fast." One of them is the average velocity, which is the ratio of the displacement : that occurs during a particular time interval to that interval^

(1-2)

The notation means that the position is at time and then at time . A common unit for is the meter per second (m/s). You may see other units in the problems, but they are always in the form of length/time.

Figure 1-4

On a graph of versus , is the slopeof the straight line that connects two particular points on the curve: one is the point that corresponds to and , and the other is the point that corresponds to and . Like displacement, has both magnitude and direction (it is another vector quantity). Its magnitude is the magnitude of the line's slope. A positive (and slope) tells us that the line slants upward to the right; a negative (and slope), that the line slants downward to the right. The average velocity always has the same sign as the displacement because in Eq. 1-2 is always positive. Figure 1-4 shows how to find for the armadillo of Fig. 1-3, for the time interval s to s. We draw the straight line that connects the point on the position curve at the beginning of the interval and the point on the curve at the end of the interval. Then we find the slope of the straight line. For the given time interval, the average velocity is

m/s.

Average speedis a different way of describing "how fast" a particle moves. Whereas the average velocity involves the particle's displacement , the average speed involves the total distance covered (for example, the number of meters moved), independent of direction; that is,

Because average speed does not include direction, it lacks any algebraic sign. Sometimes is the same (except for the absence of a sign) as . However, as is demonstrated in Sample Problem 1-1, when an object doubles back on its path, the two can be quite different.

Example 1-1

You drive a beat-up pickup truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops. Over the next 30 min, you walk another 2.0 km farther along the road to a gasoline station.

(a) What is your overall displacement from the beginning of your
drive to your arrival at the station?

Solution. Assume, for convenience, that you move in the positive direction of an axis, from a first position of to a second position of at the station. That second position must be at km. Then your displacement along the axis is the second position minus the first position. From Eq. 1-1, we have

km. (Answer)

Thus, your overall displacement is 10.4 km in the positive direction of the axis.

(b) What is the time interval from the beginning of your drive
to your arrival at the station?

Solution. We already know the time interval (= 0.50 h) for the walk, but we lack the time interval for the drive. However, we know that for the drive the displacement is 8.4 km and the average velocity is 70 km/h. A Key Idea to use here comes from Eq. 1-2: This average velocity is the ratio of the displacement for the drive to the time interval for the drive:

Rearranging and substituting data then give us

h.

So, h. (Answer)

(c) What is your average velocity from the beginning of your drive to your arrival at the station? Find it both numerically and graphically.

Solution. The Key Idea here again comes from Eq. 1-2: for the entire trip is the ratio of the displacement of 10.4 km for the entire trip to the time interval of 0.62 h for the entire trip. With Eq. 1-2, we find it is

km/h (Answer)

To find graphically, first we graph as shown in Fig. 1-5, where the beginning and arrival points on the graph are the origin and the point labeled as "Station." The Key Idea here is that your average velocity is the slope of the straight line connecting those points; that is, it is the ratio of the rise ( = 10.4 km) to the run ( = 0.62 h), which gives us = 16.8 km/h.

Fig. 1-5

(d) Suppose that to pump the gasoline, pay for it, and walk back to the truck takes you another 45 min. What is your average speed from the beginning of your drive to your return to the truck with the gasoline?

Solution.: The Key Idea here is that your average speed is the ratio of the total distance you move to the total time interval you take to make that move. The total distance is 8.4 km + 2.0 km + 2.0 km = 12.4 km. The total time interval is 0.12 h + 0.50 h + 0.75 h = 1.37 h. Thus, Eq. 2-3 gives us

km/h (Answer)

1-4 Instantaneous Velocity and Speed

You have now seen two ways to describe how fast something moves: average velocity and average speed, both of which are measured over a time interval . However, the phrase "how fast" more commonly refers to how fast a particle is moving at a given instant - and that is its instantaneous velocity(or simply velocity).

The velocity at any instant is obtained from the average velocity by shrinking the time interval closer and closer to 0. As dwindles, the average velocity approaches a limiting value, which is the velocity at that instant:

(1-4)

This equation displays two features of the instantaneous velocity . First, is the rate at which the particle's position is changing with time at a given instant; that is, is the derivative of with respect to . Second, at any instant is the slope of the particle's position-time curve at the point representing that instant. Velocity is another vector quantity and thus has an associated direction.

Speedis the magnitude of velocity; that is, speed is velocity that has been stripped of any indication of direction, either in words or via an algebraic sign. A velocity of +5 m/s and one of -5 m/s both have an associated speed of 5 m/s. The speedometer in a car measures the speed, not the velocity, because it cannot determine the direction.

Example 2-3

The position of a particle moving on an axis is given by

with in meters and in seconds. What is its velocity at s? Is the velocity constant, or is it continuously changing?

Solution. For simplicity, the units have been omitted from Eq. 1-5, but you can insert them if you like by changing the coefficients to 7.8 m, 9.2 m/s, and -2.1 m/s³. The Key Idea here is that velocity is the first derivative (with respect to time) of the position function . Thus, we write

which becomes

1-6 (Answer)

At s, the particle is moving in the negative direction of (note the minus sign) with a speed of 68 m/s. Since the quantity appears in Eq. 1-6, the velocity depends on and so is continuously changing.

Exercises

1.5 A hobbyist is testing a new model rocket engine by using it to propel a cart along a model railroad track. He determines that its motion along the -axis is described by the equation , where = 10 cm/s². Compute the instantaneous velocity of the cart at time s.

1.6 A car is initially stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by , where b= 4.0 m/s and с = 0.5 m/s².

a) Calculate the average velocity of the car for the time interval = 0to =10s.

b)Calculate the instantaneous velocity of the car at i) t =0; ii) t = 5 s; iii) t =10 s

1-5 Acceleration

When a particle's velocity changes, the particle is said to undergo acceleration (or to accelerate). For motion along an axis, the average acceleration over a time interval is

(1-7)

where the particle has velocity at time and then velocity at time . The instantaneous acceleration(or simply acceleration)is the derivative of the velocity with respect to time:

(1-8)

In words, the acceleration of a particle at any instant is the rate at which its velocity is changing at that instant. Graphically, the acceleration at any point is the slope of the curve of at that point.

We can combine Eq. 1-8 with Eq. 1-4 to write

(1-9)

In words, the acceleration of a particle at any instant is the second derivative of its position with respect to time.

A common unit of acceleration is the meter per second per second: m/(s • s) or m/s². You will see other units in the problems, but they will each be in the form of length/(time • time) or length/time². Acceleration has both magnitude and direction (it is yet another vector quantity). Its algebraic sign represents its direction on an axis just as for displacement and velocity; that is, acceleration with a positive value is in the positive direction of an axis, and acceleration with a When is constant, the derivative is zero and so also is the acceleration.

Your body reacts to accelerations (it is an accelerometer) but not to velocities (it is not a speedometer). When you are in a car traveling at 90 km/h or an airplane traveling at 900 km/h, you have no bodily awareness of the motion. However, if the car or plane quickly changes velocity, you may become keenly aware of the change, perhaps even frightened by it.

Large accelerations are sometimes expressed in terms of units, with

m/s²

Example 1-4

A particle's position on the axis of Fig. 1-1 is given by

, with in meters and in seconds.

a) Find the particle's velocity function and acceleration function .

Solution. One Key Idea is that to get the velocity function , we differentiate the position function with respect to time. Here we find

with in meters per second.

Another Key Idea is that to get the acceleration function , we differentiate the velocity function with respect to time. This gives us

, (Answer)

with in meters per second squared.

(b) Is there ever a time when ?

Solution. Setting yields

0 = -27 + 3t², which has the solution

t = ±3 s. (Answer)

Thus, the velocity is zero both 3 s before and 3 s after the clock reads 0.

(c) Describe the particle's motion for .

Solution. The Key Idea is to examine the expressions for , , and .

At , the particle is at m and is moving with a velocity of m/s - that is, in the negative direction of the axis. Its acceleration is , because just then the particle's velocity is not changing.

For s, the particle still has a negative velocity, so it continues to move in the negative direction. However, its acceleration is no longer 0 but is increasing and positive. Because the signs of the velocity and the acceleration are opposite, the particle must be slowing.

Indeed, we already know that it stops momentarily at s. Just then the particle is as far to the left of the origin in Fig. 2-1 as it will ever get. Substituting s into the expression for , we find that the particle's position just then is m. Its acceleration is still positive.

For s, the particle moves to the right on the axis. Its acceleration remains positive and grows progressively larger in magnitude. The velocity is now positive, and it too grows progressively larger in magnitude.

Exercises

1.33 A motorcycle's position is described by x = A + Bt + Ct^s, where A, B, and С are numerical constants.

a)Calculate the velocity of the cycle, as a function of time.

b) From your answer to (a), calculate the acceleration of the motorcycle as a function of time.

1.. Are rest and motion absolute or relative terms

1.02. Under what condition will the distance and displacement of a moving object will have the same magnitude?

1..03. What will be nature of graph for a uniform motion?

1.04. Can graph be a straight line parallel to time-axis?

1.05. Can graph be a straight line parallel to position-axis?

1.06. What does slope of graph represent?

1.07. Can graph have a negative slope?

1..08. Can a particle in one dimensional motion with zero speed may have non-zero velocity

1..09. Can a body have a constant velocity but a varying speed?

1..10. Can a body have a constant speed but a varying velocity? Or Is it possible for a body to be accelerated, if its speed is constant ? If it is so, give an example.

Ans. Yes. When a body moves along a circular path with/ uniform angular speed, it possesses constant speed but a varying velocity.

1..11. What does the speedometer of a car measure?

1..12. When will the relative velocity of two moving objects be zero?

1..06 Is magnitude of the displacement of an object and the total distance covered by it in certain time interval same? Explain.

1. A cyclist moves along a circular path of radius 70 m. If he completes one round in 11 s, calculate (i) total length of path, (ii) magnitude of the displacement, (iii) average speed and (iv) magnitude of average velocity.

[Ans. (i) 440 m ; (it) zero ; (iii) 40 m s^_1; (iv) zero]