Physics Study Guide/Linear motion

Kinematics

Kinematics is the description of motion. The motion of a point particle is fully described using three terms - displacement, velocity, and acceleration. For real objects (which are not mathematical points), translational kinematics describes the motion of an object's center of mass through space, while angular kinematics describes how an object rotates about its center of mass. In this section, we focus only on translational kinematics.

Displacement, velocity, and acceleration are defined as follows.

Position

Wiktionary defines "vector" as "a quantity that has both magnitude and direction, typically written as a column of scalars". That is, a number that has a direction assigned to it.

In physics, a vector often describes the motion of an object. For example, Warty the Woodchuck goes 35 feet toward a hole in the ground.

We can divide vectors into parts called "components" that each describe a part of the vector. Usually a vector is divided into x and y components.

One dimensional coordinate system

Two dimensional coordinate system

Three dimensional coordinate system

Displacement

Displacement answers the question, "Has the object moved?"

Note the $\equiv$ symbol. This symbol is a sort of "super equals" symbol, indicating that not only does ${\overrightarrow{x}}_f-{\overrightarrow{x}}_i$ EQUAL the displacement $\mathrm{\Delta }\overrightarrow{x}$, but more importantly displacement is OPERATIONALLY DEFINED by ${\overrightarrow{x}}_f-{\overrightarrow{x}}_i$.

We say that ${\overrightarrow{x}}_f-{\overrightarrow{x}}_i$ operationally defines displacement, because ${\overrightarrow{x}}_f-{\overrightarrow{x}}_i$ gives a step by step procedure for determining displacement. Namely ...

1. Measure where the object is initially.
2. Measure where the object is at some later time.
3. Determine the difference of these two position values.

Be sure to note that DISPLACEMENT is NOT the same as DISTANCE traveled.

For example, imagine traveling one time along the circumference of a circle. If you end where you started, your displacement is zero, even though you have clearly traveled some distance. In fact, displacement is an average distance traveled. On your trip along the circle, your north and south motion averaged out, as did your east and west motion.

Clearly we are losing some important information. The key to regaining this information is to use smaller displacement intervals. For example, instead of calculating your displacement for your trip along the circle in one large step, consider dividing the circle into 16 equal segments. Calculate the distance you traveled along each of these segments, and then add all your results together. Now your total traveled distance is not zero, but something approximating the circumference of the circle. Is your approximation good enough? Ultimately, that depends on the level of accuracy you need in a particular application, but luckily you can always use finer resolution. For example, we could break your trip into 32 equal segments for a better approximation.

Returning to your trip around the circle, you know the true distance is simply the circumference of the circle. The problem is that we often face a practical limitation for determining the true distance traveled. (The traveled path may have too many twists and turns, for example.) Luckily, we can always determine displacement, and by carefully choosing small enough displacement steps, we can use displacement to obtain a pretty good approximation for the true distance traveled. (The mathematics of calculus provides a formal methodology for formally estimating a "true value" through the use of successively better approximations.) In the rest of this discussion, I will replace $\mathrm{\Delta }$ with $\delta$ to indicate that small enough displacement steps have been used to provide a good enough approximation for the true distance traveled.

Velocity

[Δ, delta, upper-case Greek D, is a prefix conventionally used to denote a difference.] Velocity answers the question "Is the object moving now, and if so - how quickly?"

Once again we have an operational definition: we are told what steps to follow to calculate velocity.

Note that this is a definition for average velocity. The displacement Δx is the vector sum of the smaller displacements which it contains, and some of these may subtract out. By contrast, the distance traveled is the scalar sum of the smaller distances, all of which are non-negative (they are the magnitudes of the displacements). Thus the distance traveled can be larger than the magnitude of the displacement, as in the example of travel on a circle, above. Consequently, the average velocity may be small (or zero, or negative) while the speed is positive.

If we are careful to use very small displacement steps, so that they come pretty close to approximating the true distance traveled, then we can write the definition for INSTANTANEOUS velocity as Template:PSG/eq

[δ is the lower-case delta.] Or with the idea of limits from calculus, we have ...

[d, like Δ and δ, is merely a prefix; however, its use definitely specifies that this is a sufficiently small difference so that the error--due to stepping (instead of smoothly changing) the quantity--becomes negligible.]RLittauer (talk) 19:56, 22 December 2007 (UTC)

Acceleration

Template:PSG/eq Acceleration answers the question "Is the object's velocity changing, and if so - how quickly?"

Once again we have an operational definition. We are told what steps to follow to calculate acceleration.

Again, also note that technically we have a definition for AVERAGE acceleration. As for displacement, if we are careful to use a series of small velocity changes, then we can write the definition for INSTANTANEOUS acceleration as Template:PSG/eq

Or with the help of calculus, we have ...

Vectors

Notice that the definitions given above for displacement, velocity, and acceleration included little arrows over many of the terms. The little arrow reminds us that direction is an important part of displacement, velocity, the change in velocity, and acceleration. These quantities are VECTORS. By convention, the little arrow always points right when placed over a letter. So for example, $\overrightarrow{v}$ just reminds us that velocity is a vector, and does NOT imply that this particular velocity is rightward. Why do we need vectors? As a simple example, consider velocity. It is not enough to know how fast one is moving. We also need to know which direction we are moving. Less trivially, consider how many different ways an object could be experiencing an acceleration (a change in its velocity). Ultimately there are three distinct ways an object could accelerate.

1. The object could be speeding up.
2. The object could be slowing down.
3. The object could be traveling at constant speed, while changing its direction of motion.

(More general accelerations are simply combinations of 1 and 3 or 2 and 3).

Importantly, a change in the direction of motion is just as much an acceleration as is speeding up or slowing down. In classical mechanics, no direction is associated with time (you cannot point to next Tuesday). So the definition of ${\overrightarrow{a}}_{av}$ tells us that acceleration will point wherever the CHANGE in velocity $\mathrm{\Delta }\overrightarrow{v}$ points. Understanding that the direction of $\mathrm{\Delta }\overrightarrow{v}$ determines the direction of $\overrightarrow{a}$ leads to three non-mathematical but very powerful rules of thumb.

1. If the velocity and acceleration of an object point in the same direction, the object's speed is increasing.
2. If the velocity and acceleration of an object point in opposite directions, the object's speed is decreasing.
3. If the velocity and acceleration of an object are perpendicular to each other, the object's speed stays constant, while the object's direction of motion changes.

(Again, more general motion is simply a combination of 1 and 3 or 2 and 3.)

Using these three simple rules will dramatically help your intuition of what is happening in a particular problem. In fact, much of the first semester of college physics is simply the application of these three rules in different formats.

Equations of motion : Constant acceleration

A particle is said to move with constant acceleration if its velocity changes by equal amounts in equal intervals of time, no matter how small the intervals may be.

Since acceleration is a vector constant acceleration means that both direction and magnitude of this vector don't change during the motion. This means that average and instantaneous acceleration are equal. We can use that to derive an equation for velocity as a function of time by integrating the constant acceleration.

Giving the following equation for velocity as a function of time.

To derive the equation for position we simply integrate the equation for velocity.

Integrating again gives the equation for position.

The following are the 'Equations of Motion'. They are simple and obvious equations if you think over them for a while.

Equations of Motion

Equation	Description
$\overrightarrow{x}={\overrightarrow{x}}_0+{\overrightarrow{v}}_{0}t+\frac{\overrightarrow{a}{t}^2}{2}$	Position as a function of time
$\overrightarrow{v}={\overrightarrow{v}}_0+\overrightarrow{a}t$	Velocity as a function of time
The following equations can be derived from the two equations above by combining them and eliminating variables.
$v^2=v_0^2+2\overrightarrow{a}\cdot (\overrightarrow{x}-{\overrightarrow{x}}_0)$	Eliminating time (Very useful, see the section on Energy)
$\overrightarrow{x}={\overrightarrow{x}}_0+\frac{{\overrightarrow{v}}_{0}t+\overrightarrow{v}t}{2}$	Eliminating acceleration

Key to Symbols

Symbol	Description
$\overrightarrow{v}$	velocity at time t
$\overrightarrow{v_0}$	initial velocity
$\overrightarrow{a}$	acceleration (constant)
$t$	time taken during the motion
$\overrightarrow{x}$	position at time t
$\overrightarrow{x_0}$	initial position

Acceleration in One Dimension

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Acceleration in Two Dimensions

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Acceleration in Three Dimensions

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