Imagine two strings tied to a box.
Case 1: Two strings are pulled with the same $u$ velocity. The box will also move with velocity $u$.
Case 2 : Tension along $\text{a}$ string is $T$. Therefore total force acting on the box is $T+T=2T$. (Box is accelerating)
I think my problem is obvious. Both velocity and force are vectors. But why we can not get the velocity of the box in the first case as $u+u=2u$? (This is obviously wrong, but why?)
I'll start my answer by departing from the specific example, to provide a more general answer. In the end, I'll summarize how the general discussion applies to the specific example.
Let's focus on your observation that
Both velocity and force are vectors.
True. Both are vectors, and an intrinsic characterization of vectors is that there is a sum of vectors. However, although the vector addition is a well definite mathematical concept (in essence, the parallelogram rule), the application of vectors in physics cannot avoid the additional step of carefully identifying the physical meaning of the sum.
When we identify the physical entity force with vectors, we implicitly or explicitly have to provide an operative meaning of all the vector operations (sum and multiplication by a scalar). In classical mechanics, if we identify the presence of a force ${\bf F}$ by the resulting acceleration of a test particle, the sum of two forces applied to the same particle and product of a force by a scalar, are directly related to the corresponding addition of accelerations and multiplication of acceleration by a scalar.
Notice that an important ingredient of the force addition concept is adding only forces applied to the same body. Missing that, we would go into trouble if we try to sum an action-reaction pair of forces. In a more mathematical flavor, we could say that forces on different bodies are in different vector spaces and therefore cannot be summed.
A similar discussion can be done on velocities. Displacements of a point-like object in a time $\Delta t$, can be represented by vectors. How do we know that? We simply define the sum of two displacements of the same body as the resulting displacement. With this definition, it is a non-trivial physical finding that the order of the two displacements does not matter (sum is commutative), that there is a zero-displacement. There is an opposite displacement for every displacement such that the sum of both is equivalent to the zero displacement. Moreover, it is possible to define multiplication by a scalar, by using displacements in the same direction. Such a multiplication fulfills all the corresponding axioms in the definition of a vector space.
The key point is that the sum of displacements as vectors has the physical meaning of combining different displacements of the same body. Whatever can be said about displacements, can be said about velocities, of course.
To summarize, what can or cannot be done when summing entities called velocities or forces depends on the physical meaning we give to the mathematical concepts. It is not enough to have vector quantities to sum them without analyzing what kind of vectors are.
Let's now come to the example. Forces on the same body can be summed and this would result in an acceleration which is the sum of the accelerations present if only one of the forces at a time were present. Summing velocities of two different points of the same rigid body is meaningless because the body (the box) position is identified by one point only.
Notice that clarifying the concepts behind vector addition of velocities is an important prerequisite to avoid confusion with the laws of transformation of velocities in different reference frames in Relativity.
Two strings are pulled with same velocity $u$. The box will also move with velocity $u$.
You don't "pull" a box with a velocity. You can pull it by applying a force that results in a change of velocity. Imagine you have two forces acting on the box, one that gives it a velocity $\vec v$ and the other gives it a velocity $\vec w$. Then it is okay to say that the resulting velocity is $\vec v +\vec w$.
Tension along a string is $T$. Therefore total force acting on the box is $T+T=2T$.(Box is accelerating)
In this case, you are applying two forces to the box, and so they add to give you a resultant force. The first case doesn't really make sense physically, but the second case does.
Both velocity and force are vectors
Yes they are, and mathematically we can add any two vectors, but when adding vectors that represent physical quantities, we need to be sure what these physical quantities represent.
In your first example, you cannot add different velocities at different points on the same body since its velocity is defined by the translational motion of one point (usually the center) of the body (assuming the body is rigid).
The second example with forces works fine since the net force on an object is the vector sum of all forces acting on it, as you have done.
You are walking down the street with your friend. Now you hold hands. Are you now moving twice as fast?
If addition of velocity worked like that, you would get total nonsense. Whenever two moving objects would stick to each other, they would move faster.
Another, more mathematical way of looking at it. The energy of an object of mass $m$ moving at velocity $v$ is $$E = \frac12 mv^2.$$ Now mentally imagine dividing this object in two objects of masses $m_1$ and $m_2$ (like person = body + head). We obviously have $$m = m_1+m_2.$$ Now imagine these two parts are moving with velocities $v_1$ and $v_2$ respectively. But since mentally dividing an object cannot change its energy, we must have $$\frac12 mv^2 = \frac12\left(m_1v_1^2 + m_2v_2^2\right).$$ Solving this will lead you to $$v=v_1=v_2$$
Sorry for my poor english. French is my native language.
To define a vector, it is necessary to specify the vector space on which it is defined. In general, for a manifold, we have a tangent vector space at each point. In classical physics, space has an affine flat space structure and we define at each point a tangent vector space which, for an affine flat space, can all be identified with each other.
It is in this tangeant vector space that the addition of two vectors is defined. One thus defines at each point the vector space of the displacements. The vector sum of two displacements from a point is a displacement. By dividing by time, we go to velocity vectors.
On the other hand, there are difficulties in defining the derivative of the velocity : we compare vectors at different points. To be able to do this, we have to define a parallel transport and connection that allows a vector to be transported from one point to another. It is very easy in the case of the affine space of classical physics. More complicated in a variety: it is necessary to introduce a covariant derivative.
So even the addition of two forces at different points is not a simple consequence of the structure of vector space. We have to transport the vectors. And this is only simple for an affine flat space.
This is an insightful question.
Just like not all things with legs are tables, not all things represented by vectors (in a mathematical sense) are the same.
In classical mechanics, there are two classes of vector quantities, each with some common properties. The nomenclature below isn't standardized, as different authors have used different names for the same concepts below.
Axis Vector - A unique vector that conveys the direction and magnitude of a quantity belonging to the body that represents a line (or axis) in space.
Some examples on a rigid body are
The momentum vector is a single vector describing the translational momentum state of a rigid body. Regardless of how a body rotates, it is always defined as $$\boldsymbol{p} = m\, \boldsymbol{v}_{\rm C}$$ where C is the center of mass. The line associated with momentum is called the axis of percussion.
The force vector is a single vector describing the loading a body is under. Force is the time derivative of momentum and the total load is thus $$ \boldsymbol{F} = m \, \boldsymbol{a}_C$$ The line associated with force is called the line of action.
The Rotational velocity vector is a common quantity shared by all particles of a body. Any point on the body (or the general extended frame) is said to rotate by $\boldsymbol{\omega}$ relative to each other point. The line associated with rotation is called the rotation axis.
Moment Vector - A vector that varies by location and is defined by taking the moment of an axis vector. This requires the right-hand rule convention in the form of a cross product to define the direction of the vector. This defines a vector field around the lines mentioned above. A vector field is a vector that changes direction and magnitude by location.
Some examples are at some arbitrary point A are:
Now for the vector algebra part of the question. How do we add two forces or two velocities and how does this process differ.
The result is the vector addition of the axis vectors (Forces) and the moment vectors (Torques) component by component.
To describe the kinematics of a body relative to another body you need to add up both the axis vectors (rotational velocity) and the moment vectors (translational velocity) expressed at a common point, just like the forces above need to be added at a common point.
The geometry of this situation leads to the relative centre of rotation theorem.
It is worth reading the following article from 1901
Additionally, read this answer here about the nature of torque and the moment vectors that are defined using the cross product.
In summary, the common quantities in mechanics are interpreted as follows $$ \begin{array}{r|l|l} \text{concept} & \text{value} & \text{moment}\\ \hline \text{rotation axis} & \text{rot. velocity}, \boldsymbol{\omega} & \text{velocity}, \boldsymbol{v} = \boldsymbol{r}\times \boldsymbol{\omega} \\ \text{line of action} & \text{force}, \boldsymbol{F} & \text{torque}, \boldsymbol{\tau} = \boldsymbol{r} \times \boldsymbol{F} \\ \text{axis of percussion} & \text{momentum}, \boldsymbol{p} & \text{ang. momentum}, \boldsymbol{L} = \boldsymbol{r} \times \boldsymbol{p} \end{array} $$ The stuff under the value column are fundamental quantities that give us the magnitude of something (as well as the direction). The stuff under the moment column are secondary quantities that depend on where they are measured and give use the relative location of the fundamental quantities. Hence the terms torque = moment of force, velocity = moment of rotation and angular momentum = moment of momentum. All that means is that these quantities are $\boldsymbol{r} \times \text{(something fundamental)}$ and they describe the moment arm to this something.
The fact that velocity and force are vectors is only secondary here. The true distinction here is whether the quantities in question are intensive or extensive, that is, how they scale with the size of the system.
With scalar values, we could ask exactly the same thing about temperature and mass. Consider two identical objects with temperature $T$ and mass $m$. When put together, their temperature will not increase to $T$, but will still remain at $T$, because temperature is an intensive property. However, mass is an extensive property, so the combined mass will increase to $2m$.
Combining two unequal extensive properties reduces to a sum, while with intensive properties it is a weighted average over some extensive property of a system: in this example it would be the total heat capacity of each of the bodies.
In case (1) constant velocity means the acceleration is zero. In the absence of gravity, there is no net force. Only relative velocities add.
