Physics Study Guide/Linear motion

Kinematics is the description of motion. The motion of a point particle is fully described using three terms - position, 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 centre of mass. In this section, we focus only on translational kinematics. Position, displacement, velocity, and acceleration are defined as follows.

Position

"Position" is a relative term that describes the location of an object RELATIVE to some chosen stationary point that is usually described as the "origin".

A vector is 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 10 meters towards a hole in the ground.

We can divide vectors into parts called "components", of which the vector is a sum. For example, a two-dimensional vector is divided into x and y components.

One dimensional coordinate system

Two dimensional coordinate system

Three dimensional coordinate system

Displacement

$\Delta {\vec {x}}\equiv {\vec {x}}_{f}-{\vec {x}}_{i}\,$

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 ${\vec {x}}_{f}-{\vec {x}}_{i}$ EQUAL the displacement $\Delta {\vec {x}}$, but more importantly displacement is operationally defined by ${\vec {x}}_{f}-{\vec {x}}_{i}$.

We say that ${\vec {x}}_{f}-{\vec {x}}_{i}$ operationally defines displacement, because ${\vec {x}}_{f}-{\vec {x}}_{i}$ 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 between these two position values.

Be sure to note that displacement is not the same as distance travelled.

For example, imagine travelling one time along the circumference of a circle. If you end where you started, your displacement is zero, even though you have clearly travelled some distance. In fact, displacement is an average distance travelled. 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 travelled along each of these segments, and then add all your results together. Now your total travelled 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 travelled. (The travelled 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 travelled. (The mathematics of calculus provides a formal methodology for estimating a "true value" through the use of successively better approximations.) In the rest of this discussion, I will replace $\Delta$ with $\delta$ to indicate that small enough displacement steps have been used to provide a good enough approximation for the true distance travelled.

Velocity

${\vec {v}}_{av}\equiv {\frac {{\vec {x_{f}}}-{\vec {x_{i}}}}{t_{f}-t_{i}}}\equiv {\frac {\Delta {\vec {x}}}{\Delta t}}$

[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 travelled is the scalar sum of the smaller distances, all of which are non-negative (they are the magnitudes of the displacements). Thus the distance travelled 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 travelled, then we can write the definition for instantaneous velocity as

${\vec {v}}_{inst}\equiv {\frac {\vec {\delta x}}{\delta t}}$

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

${\vec {v}}_{inst}\equiv {\frac {d{\vec {x}}}{dt}}$

[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.]

Acceleration

${\vec {a}}_{av}\equiv {\frac {{\vec {v_{f}}}-{\vec {v_{i}}}}{t_{f}-t_{i}}}\equiv {\frac {\Delta {\vec {v}}}{\Delta t}}$

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:

${\vec {a}}_{inst}\equiv {\frac {\delta {\vec {v}}}{\delta t}}$

Or with the help of calculus, we have:

${\vec {a}}_{inst}\equiv {\frac {d{\vec {v}}}{dt}}={\frac {d^{2}{\vec {x}}}{dt^{2}}}$

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, and acceleration. These quantities are vectors. By convention, the little arrow always points right when placed over a letter. So for example, ${\vec {v}}$ 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 ${\vec {a}}_{av}$ tells us that acceleration will point wherever the change in velocity $\Delta {\vec {v}}$ points.

Understanding that the direction of $\Delta {\vec {v}}$ determines the direction of ${\vec {a}}$ 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 initial speed stays constant (in that initial direction), while the speed of the object in the direction of the acceleration increases. Think of a bullet fired horizontally in a vertical gravitational field. Since velocity in the one direction remains constant, and the velocity in the other direction increases, the overall velocity (absolute velocity) also increases.

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

${\frac {d{\vec {a}}}{dt}}=0\ \mathrm {\frac {m}{s^{3}}}$

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.

${\boldsymbol {v}}(t)={\boldsymbol {v}}(0)+\int \limits _{0}^{t}{\boldsymbol {a}}\ dt$

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

${\boldsymbol {v}}(t)={\boldsymbol {v}}_{0}+{\boldsymbol {a}}t$

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

${\boldsymbol {x}}(t)={\boldsymbol {x}}(0)+\int \limits _{0}^{t}{\boldsymbol {v}}(t)\ dt$

Integrating again gives the equation for position.

${\boldsymbol {x}}(t)={\boldsymbol {x}}_{0}+{\boldsymbol {v}}_{0}t+{\frac {1}{2}}{\boldsymbol {a}}t^{2}$

The following are the equations of motion:

Equations of Motion

Equation	Description
${\vec {x}}={\vec {x}}_{0}+{\vec {v}}_{0}t+{\frac {{\vec {a}}t^{2}}{2}}\$	Position as a function of time
${\vec {v}}={\vec {v}}_{0}+{\vec {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{\vec {a}}\cdot ({\vec {x}}-{\vec {x}}_{0})\$	Eliminating time (very useful, see the section on Energy)
${\vec {x}}={\vec {x}}_{0}+{\frac {{\vec {v}}_{0}t+{\vec {v}}t}{2}}$	Eliminating acceleration

Symbols

Symbol	Description
${\vec {v}}$	velocity (at time t)
${\vec {v_{0}}}$	initial velocity
${\vec {a}}$	(constant) acceleration
$t\$	time (taken during the motion)
${\vec {x}}$	position (at time t)
${\vec {x_{0}}}$	initial position

Acceleration in Two Dimensions

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Acceleration in Three Dimensions

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What does force in motion mean?

Force means strength and power. Motion means movement. That’s why we need forces and motions in our life. We need calculation when we want to know how fast things go, travel and other things which have force and motion. ...

How do we calculate the speed?

If you want to calculate the average speed, distance travelled or time taken you need to use this formula and remember it:

${speed}={\frac {distance}{time\ taken}}$

This is an easy formula to use, you can find the distance travelled, time taken or average speed, you need at least 2 values to find the whole answer.

Is velocity the same thing as speed?

Velocity is a vector quantity that refers to "the rate at which an object changes its position", whereas speed is a scalar quantity, which cannot be negative. Imagine a kid moving rapidly, one step forward and one step back, always returning to the original starting position. While this might result in a frenzy activity, it would result in a zero velocity, because the kid always returns to the original position, the motion would never result in a change in position, in other words $\Delta {\vec {x}}$ would be zero.

Speed is measured in the same physical units of measurement as velocity, but does not contain an element of direction. Speed is thus the magnitude component of velocity. Velocity contains both the magnitude and direction components. You can think of velocity as the displacement/duration, whereas speed can be though as distance/duration.

Acceleration

When a car is speeding up we say that it is accelerating, when it slows down we say it is decelerating.

How do we calculate it?

When we want to calculate it, the method goes like that: A lorry driver brakes hard, and slows from 25 m/s to 5 m/s in 5 seconds. What was the vehicle's acceleration?

${acceleration}={\frac {change\ in\ velocity}{time\ taken}}={\frac {5-25}{5}}={\frac {-20}{5}}=-4\ ms^{-2}$

What is initial velocity and final velocity? Initial velocity is the beginning before motion starts or in the middle of the motion, final velocity is when the motion stops.

There is another way to calculate it and it is like that This equations which are written is the primary ones, which means that when you don’t have lets say final velocity, how will you calculate the equation?

This is the way you are going to calculate.

Observing motion

When you want to know how fast an athletic person is running, what you need is a stopwatch in your hand, then when the person starts to run, you start the stopwatch and when the person who is sprinting stops at the end point, you stop the watch and see how fast he ran, and if you want to see if the athlete is wasting his energy, while he is running look at his movement, and you will know by that if he is wasting his energy or not.

This athletic person is running, and while he is running the scientist could know if he was wasting his energy if they want by the stop watch and looking at his momentum.

Measuring acceleration

Take a slope, a trolley, some tapes and a stop watch, then put the tapes on the slope and take the trolley on the slope, and the stopwatch in your hand, as soon as you release the trolley, start timing the trolley at how fast it will move, when the trolley stops at the end then stop the timing. After wards, after seeing the timing , record it, then you let the slope a little bit high, and you will see, how little by little it will decelerate.

Newton

Isaac Newton was an English physicist, mathematician (described in his own day as a "natural philosopher") , astronomer and alchemist. Newton is one of the most influential scientists of all time, and he is known, among other things, for contributing to development of classical mechanics and for inventing, independently from Gottfried Leibniz, calculus.

Newton's laws of motion

Newton is also known by his three laws of motion, which describe the relationship between a body and the forces acting upon it, and its motion in response to said forces.

1. First Law (also known as the law of inertia) states that every body continues in its state of rest or state of uniform motion unless compelled to change that state by being subject to an external force. The moment of inertia is defined as the tendency of matter to resist any change in its state of motion or state of rest.
2. Second Law The vector sum of the external forces $F$ on an object is equal to the mass $m$ of that object multiplied by the acceleration vector $a$ of the object, or algebraically $F=ma$.
3. Third Law states that when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

Symbols

Some useful symbols seen and that we will see:

Name	Symbol
Distance travelled	$d$ or $s$
Force	$F$
Velocity	${\vec {v}}$
Initial velocity	${\vec {v}}_{0}$ or ${\vec {v}}_{i}$
Final velocity	${\vec {v}}_{f}$
Change in velocity	$\Delta {\vec {v}}$
Acceleration	${\vec {a}}$
Mass	$m$
Newton	$N$
Gravity	$G$
Weight	$W$

Force

A force is any interaction that tends to change the motion of an object. In other words, a force can cause an object with mass to change its velocity. Force can also be described by intuitive concepts such as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol $F$.

How to calculate the force?

When we want to calculate the force, and we have the mass and acceleration, we can simply use the simple formula stated in the Newton's second law above, that is $F=ma$, where $m$ is the mass (or the amount of matter in a body), and $a$ is the acceleration. Note that the Newton’s second law is defined as a numerical measure of inertia.

What is inertia?

Inertia is the tendency of a body to maintain its state of rest or uniform motion, unless acted upon by an external force.

Robert Hooke

Robert Hooke was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work.

Hooke's law

Hooke's law is a principle of physics that states that the force $F$ needed to extend or compress a spring by some distance $X$ is proportional to that distance, or algebraically $F=-kX$, where $k$ is a constant factor characteristic of the spring, its stiffness.