Physical Quantities Sign Convention

Question

I see that almost all physical quantities carry signs. But the confusion I have is what they really mean.

1. Does negative velocity mean decreasing velocity or velocity in the opposite direction?

2. Does negative acceleration imply retardation or acceleration in the opposite direction?

3. Similarly, what does negative force and negative work signify?

Also, my textbook says

"When a body falls freely under the action of gravity, its velocity increases, and the value of $g$ is taken positive. When a body is thrown vertically upward, its velocity decreases, and the value of $g$ is taken as negative. "

But I see that in the same textbook, the signs of $g$ are chosen in contradiction to the above statements. Is there an easy way to choose the sign of $g$?

Answer 1

Other answers have answered the direct questions in the OP. Here, I'd like to expand on the general question a little bit and talk about the fact that negative signs mean different things in different contexts in physics.

There are a lot of different meanings of negative signs in physics, and one can often tell an expert from a novice by having them look at an equation and "tell the story" of the equation by explaining where each negative sign comes from. Here are a handful of "meanings" of negative signs in physics.

(For more info, here is a peer-reviewed paper coming out of the PER community entitled "Framework for the natures of negativity in introductory physics".)

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• Directions of vectors. Suppose the velocity vector is written as $\vec{v} = -(3~\mathrm{m/s})\hat{i}$. The negative sign here indicates that the velocity is in the negative-$x$ direction, so the object is traveling in the negative $x$-direction. See the next note.

• Reversing the direction of a vector. Newton's 3rd Law states that for every force $\vec{F}_{\textrm{A on B}}$ exerted by object A on object B, there is a force $\vec{F}_{\textrm{B on A}}$ of the same type exerted by object B on object A that satisfies $\vec{F}_{\textrm{B on A}}= - \vec{F}_{\textrm{A on B}}$. Here, the negative sign doesn't mean that the second force "points in the negative direction". Rather, the negative sign means that the second force points in exactly the opposite direction as the first force. In other words, attaching a negative sign to a vector reverses the direction of the vector.

Aside: The phrase "a vector is negative" isn't a meaningful statement. Rather, the negative signs must be interpreted in terms of the statements above, where the negative sign is chosen to specify the direction of the vector relative to another vector (or relative to a basis vector like $\hat{i}$, which is a unit vector that points in the positive $x$-direction).

• Change in a quantity over time. Here, the negative sign arises because we are looking at how a quantity changes. If an object is at position $\vec{x}_i$ at the initial time $t_i$ and the at the position $\vec{x}_f$ some later time $t_f$, then the displacement, or change in position is $\Delta \vec{x} = \vec{x}_f - \vec{x}_i$. The negative sign isn't indicating a direction at all; I mean, it can't, because it's attached to the position vector of an object at two different times. Instead, it's there showing an operation between two vectors that spits out the change in the quantity between the two times. This comes up over and over again: change in position is related to velocity; change in velocity is related to acceleration; change in temperature is related to the amount of heat flow into an object, etc.

• Direction: not in space, but rather indicating "into" or "out of". The work done on a particle by a (constant) net force $\vec{F}_{\textrm{net}}$ while the particle undergoes a displacement $\Delta \vec{x}$ is $W = \vec{F}_{\textrm{net}}\cdot\vec{x}$. The basic work-energy principle states that $\Delta K = W$, i.e., the change in kinetic energy of the particle is equal to the work done on it by the net force on it.

  If $\Delta K >0$, the object speeds up, i.e., its kinetic energy increases. If $\Delta K<0$, then the object slows down, i.e., it's kinetic energy decreases. More carefully, $K_f - K_i = \Delta K$. Now, suppose that the kinetic energy decreases by 5 J, which means that it must have been that $W = - 5$ J. What does this negative sign mean? It means that the object has lost 5 J of energy, i.e., that there was a flow of energy out of the system. Thus, $W$ being negative indicates that energy flowed out of the system, while $W$ being positive indicates that energy flowe into the system.

Aside: The negative sign that comes out is related to the relative directions of the force and displacement.

• Mathematical operations. Sometimes, negative signs appear just because you did some mathematical manipulations. For instance, let's suppose you were solving for the acceleration using a kinematic equation, going from $$\Delta x = v_0\Delta t +\frac{1}{2}a\Delta t$$ to $$a=2(\Delta x - v_0\Delta t)/(\Delta t)^2$$. Here, the negative sign just appeared because we subtracted $v_0\Delta t$ from both sides as part of the algebra we did to solve for $a$. Thus, it doesn't really have a direct meaning.

Finally, there are more. Here's an example which you'll see when you get to electromagnetism.

• Opposite "types". There are two types of electric charge, "positive" and "negative". An electron, for instance, has a charge of $-1.6\times 10^{-19}$ coulombs. This negative sign doesn't indicate direction, or mathematical manipulation, or a change, or anything. It's just there to tell us what type of charge it is. (Now, it's a convenient choice, mathematically, because it makes the math "nicer" later on: that negative sign, when showing up in the expression for the force a negative charge feels due to an electric field, makes the direction of the force work out correctly in the math.)

Answer 2

For a vector quantity $u\,\hat u$, if the component $u$ is negative it means that the vector is in the $\hat u'= -\hat u$ direction, ie in the opposite direction to $\hat u$.

Does negative velocity mean decreasing velocity or velocity in the opposite direction?
Negative velocity mean motion in the opposite direction to that when the velocity is positive.

Does negative acceleration imply retardation or acceleration in the opposite direction?
Negative acceleration is changing velocity in the opposite direction to that when the acceleration is positive.
The word retardation should be avoided as it is often taken to imply a decreasing speed which is not the case if with negative acceleration of $-2 \hat i= 2 (-\hat i)$ the velocity changes from $-5\hat i$ to $-10 \hat i$, ie the speed having increased from $5$ to $10$.

What does negative force $\dots$ signify? Negative force mean that the direction of the force is in the opposite direction to direction of a positive force.

What does $\dots$ negative work signify?
Work is a scalar and so its value resides on the number line ranging from $-\infty$ to $+\infty$.
Positive work is done when the component of a force is in the same directions as the displacement of the force and negative work is done when the component of a force is in the opposite directions to the displacement of the force

Answer 3

If quantity $\vec q$ is a vector (like velocity, acceleration, force), then quantity $- \vec q$ is simply same vector $\vec q$ scaled by $-1$, so that now $$ -\vec q = \left[ \begin{matrix} -q_x\\ -q_y\\ -q_z \end{matrix} \right]$$ column vector represents a reversed vector. You can see this easily if you draw some vector in $x,y$ plane for example, then invert it's components and see the result directly. Like here :

vector $(5;2)$ is inverted version of vector $(-5;-2)$ or vice-versa. So to say negative vector quantity makes mirror reflection of positive vector quantity about $0$ point.

Now if some quantity is increasing or decreasing depends solely on a coordinate choice. So depending on how you choose $y$ coordinate major (positive) axis,- going down or up,- you will get ball thrown upwards speed balance equation at the terminal point $v_0-gt=0$ or $-v_0+gt=0$. These both forms are mathematically identical, since nobody can argue which coordinate system is best.

About retardation,- it's all relative. Imagine a rocket which already goes at some initial velocity $\vec v_0$, now it turns-on it's engines and starts to propel in reversed direction. Until it reaches zero speed you can name it "retardation", but if rocket will continue same trust,- deceleration will become acceleration. Hence deceleration is same acceleration, but towards $-x$ axis end instead of $+x$ axis end. So to say we have a vector and your task is to decide where "it is shooting at", i.e. choosing right and suitable coordinate system.

In case quantity is scalar and has differential form,- like for example decay rate for radioactive isotopes, $A=-dN/dt$. We can conclude from this if differential is increasing or decreasing. Here for example since number of radioactive isotopes decreases over time $dN$ must be negative. Multiplying it by $-1$ we get positive decay rate, which means average number of atoms decayed over unit time period.

HTH!

Answer 4

One thing is a sign. Another thing is an operator.

• The negative sign means the value/vector/quantity in "the opposite direction", as you suggest.
• The subtraction operator causes reduction in a value under arithmetic subtraction.

Now, it turns out that the effects of both are perfectly equivalent, mathematically. I.e., a vector subtracted from another vector is equal to the opposite vector (with a negative sign) being added. Of that reason we can allow ourselves to use the same symbol for both the sign and the operator - the minus symbol. Same story for the plus symbol, representing both a positive sign and the additive operator.

When choosing the sign of a quantity, you must keep in mind that it is nothing but a human invention. Nature has no signs. A negative sign in itself means nothing in real life. It is only a mathematical indicator, and nothing more. And what does it indicate? It indicates whether a quantity is "along with" an arbitrarily chosen positive orientation.

• A positive orientation can be a physical direction, given by an axis. If you choose an upwards axis, then the gravitational acceleration, which obviously is downwards, should be negative. To indicate that it is opposite to the chosen positive orientation. If you choose a Dow wards axis, then its sign should be positive. The sign clearly has no real-life meaning, apart from indicating directionality according to a chosen axis.

• A positive orientation can also be in vs. out (think thermal energy streaming in or out of a window) or absorbed vs released (think of heat) or similar.

In general, in any binary scenario we can indicate direction by choosing a positive orientation and then using signs. Bottomline is that such sign can never stand alone without a chosen reference to define the positive orientation. But don't confuse this with the subtraction/addition operators which are not about directionality. Although the effects mathematically are identical.

Answer 5

Negative velocity means it is moving in the opposite direction, however negative acceleration just signifies that its velocity is decreasing at the given rate, to give some examples:

1. If something is going -3 meters per second it is moving in the negative direction by 3 meters every second.

2. If something is decelerating by 3 meters per second squared and it is moving with a velocity of 15 meters per second its velocity will be at zero meters per second after 5 seconds with no outside forces, after 6 seconds if no other forces are applied it will be at -3 meters per second and therefore will be moving in the negative direction.

3. Negative force is any force that is moving against the primary force that you are tracking such as friction going against motion. Negative work likewise is when work is being done when displacement is going against a force, for example if a rocket is launching into space with an acceleration of 1000 meters per second squared and gravity is pulling down with a force of 10 meters per second squared then the rocket will go in the positive direction but the force that is pulling down will cause negative work because the displacement is positive and the force is negative and multiplying the two will give you a negative number.

Answer 6

Velocity is a vector, so minus means head of the vector arrow in the direction of decreasing axis values.

Similarly for acceleration and force. But for something to accelerate oppositely still having positive velocity, it needs to decelerate first through application of force in the opposite direction.

Answer 7

1. Any change in either the magnitude or direction of velocity is acceleration. Since velocity is a vector $\vec v$ its negative is a vector pointing in the opposite direction.

2. They are often alternatively used.

3. It’s the same for force as it is for velocity. Work on the other hand is the dot product of force and the displacement of the point of application of the force. Since force and displacement are both vectors, work is a scalar quantity. The work is positive when the direction of the force is the same as the displacement. It results in the source of the force transferring energy to what is being displaced. The work is negative when the direction of the force is the opposite to the direction of the displacement. It results in energy being taken away from what is being displaced.

Hope this helps.