Intuition behind $p = mv$ and $F = ma$ I am only familiar with physics at an elementary level; and while that is true, it would be nice if I could clear up some clashes with intuition I'm having before progressing further. What is the reason for Isaac Newton defining force = $ma$ and momentum = $mv$?
I tried, perhaps, tackling this myself. I thought: maybe once it was realised that the quantity obtained by multiplying the mass of an object by its velocity stays constant throughout its motion, it was thought to be fair to name that quantity; hence momentum is defined as mv. But I still couldn't get an intuition on force. I can see how it's associated with the variables $m$ and $a$, but don't see why the relation is as it is.
Since I am only familiar with physics at an elementary level, it would be also appropriate to comment that this answer to this question would only be answered by executing higher degree knowledges.
 A: (a) The faster a body is moving the harder it is to bring it to rest, and the more mass a body has the harder it is to bring it to rest (from a given speed). To be more precise, a body of given mass travelling at speed 2v needs the same force applied for twice the time to bring it to rest as the same body moving at speed v. And a body of mass 2m travelling at speed $v$ needs the same force applied for twice the time to bring it to rest as a body of mass m moving at speed $v$. So a body of mass 2m travelling at speed 2v needs the same force applied for 4 times the time as a body of mass m and speed v, and so on. Hence mv seems to be a good way to quantify a body's momentum.
We also assign momentum a direction – that in which the body is moving. This is effected by defining a body's momentum as $m\mathbf v$, in which $\mathbf v$ is the body's velocity. As a vector quantity – see Claudio Saspinski's answer – momentum has remarkable properties.
(b) I expect that Newton started out with the same intuitive understanding of force that we all have. We can feel forces and we can exert forces, perhaps using our hands. Scientists of his era realised that forces could be measured: for example, Hooke's law (1660) makes use of a quantitative concept of force. Newton had the brilliant idea that whenever a body's momentum changed there was always a (net) force acting on it. Even when a ball rolling on a horizontal surface seemed to slow down and stop all by itself, or when a planet in orbit continuously changed its direction of motion. Putting these ideas together (or so we might surmise) Newton proposed that the rate at which a body's momentum changed was simply proportional to the size of the (net) force acting on it. Moreover, the direction of the momentum change was same as the direction in which the force acted. Mathematically, if a constant net force $\mathbf F$ acts on a body of mass $m$, changing its velocity from $\mathbf v_1$ to $\mathbf v_2$ in time $t$, then
$$\mathbf F \propto \frac{m\mathbf v_2 - m\mathbf v_1}t$$
We can factor out $m$ and write this as
$$\mathbf F \propto m\frac{\mathbf v_2 - \mathbf v_1}t\ \ \ \ \ \ \text{that is}\ \ \ \ \ \ \mathbf F \propto m\mathbf a$$
In the SI the units of force are chosen so that the proportionality becomes an equality!
[The force need not be constant; this restriction was made in the argument above just so that calculus notation could be avoided]
A: It's best to think of the two relationships in terms of Newton's 1st and 2nd laws taken together.
Newton's 1st law states, in effect: "An object moving at constant speed in a straight line continues to move at constant speed in a straight line unless acted upon by a net external force". That described motion is called the linear momentum of the object, or $\vec p=m\vec v$. The objects "wants" to continue its motion in a straight due to the inertia of its mass.
The linear motion (momentum) is conserved unless the object is acted upon by a net external force $\vec {F}_{net}$, in which case its motion changes (accelerates) according to Newton's 2nd law, where $\vec {F}_{net}=m\frac{d\vec v}{dt}=m\vec a$.
Hope this helps.
A: They are convenient definitions due to the result of some experiments.
Several moving particles without external forces applied can collide, changing their velocities, but the total momentum ($m_1\mathbf{v_1} + m_2\mathbf{v_2} + ... + m_n\mathbf{v_n}$) doesn't change. So, it is a remarkable quantity.
When a single force (measured by the displacement of a spring) is applied to a particle, that displacement is approximately proportional to the product of its mass by its acceleration. We can define net force as $\mathbf F = m\mathbf a$, and any small discrepancy beyond the experimental precision is now a lack of linearity of the spring ($F \approx kx$).
A: I'd say that these two are quite different one from the other:

*

*$\mathbf{p} = m \mathbf{v}$ is the definition of the momentum of a particle. In other words "momentum is defined as mass times velocity"


*$\mathbf{f} = m \mathbf{a}$ is not a definition, but a principle of Newton dynamics, namely the principle that links together inertia, kinematics and actions acting on a system. In other words "force acting on a system equals (while it is not) the product of mass and acceleration (or better, the time derivative of the momentum)".
