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$$F = ma$$

This was never an intuitive equation. A much more intuitive equation would be:

$$F = m \cdot velocity$$

When you punch someone, the force exerted seems to be proportional to velocity.

Is there an intuitive way to understand $F = ma$, or to understand why $F\neq m\cdot velocity$?

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    $\begingroup$ It's an experimental fact: the intuition is that since the acceleration depends quadratically on the time, it can be nothing but proportional to the force (see free falling objects for instance). $\endgroup$ – gented Mar 20 '17 at 11:14
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    $\begingroup$ because punching isn't primarily about force, but energy and momentum; your intuition isn't entirely wrong, you're just mislead by terminology $\endgroup$ – Christoph Mar 20 '17 at 11:18
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    $\begingroup$ Why would the latter more "intuitive"? What are you defining force to be here? $\endgroup$ – ACuriousMind♦ Mar 20 '17 at 12:07
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    $\begingroup$ Why do these things always seem to involve punching people in the face? $\endgroup$ – Jim Mar 20 '17 at 13:13
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    $\begingroup$ What you are saying is that, for you, momentum and impulse are more "intuitive" concepts than force. That's fine, but if you insist on calling something "force" when everyone else calls it a different name, that probably won't end well if you want to study physics seriously. $\endgroup$ – alephzero Mar 20 '17 at 13:45
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Punching someone in the face involves changing the velocity of your fist. If your fist goes from very fast to not moving, this is greater acceleration than not-so-fast to not moving (provided this change in speed occurs over the same time interval). So the faster fist does provide the largest bunch but its acceleration from punch speed to rest is greatest.

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What is more difficult: keep some heavy object in hands when you are standing on ground or when you are standing inside a moving train? Actually there is no difference. You do not need to push this heavy object forward in a direction of a train's velocity to keep it moving.

More than that, if train's velocity decreases, the object "tries" to move forward from your hands. You have to apply force in a direction opposite to the direction of the velocity.

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The force that you exert on the person depends on the rate of change of the velocity of your fist.
If there was no change in the velocity of your fist the person would not feel a force.
For example if the person was moving away from your fist the force that you exerted on that person would be less - boxers do that all the time until they walk into a punch when your fist might actually rebound causing a greater change in the velocity.

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When you punch someone your hand is not moving with a constant velocity.

It has a constant velocity and then has negative acceleration as it slows down when you are hitting them. This acceleration is where the force comes from.

The faster it is, the more it must accelerate to slow down. This is why a faster punch has more force.

Another way to look at it is it has more total energy, so it can apply greater force when stopped.

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I will move away from the undoubted satisfaction of punching (which is sometimes justified, in my opinion, but also dangerous as they might punch you back) to switching to the example of the acceleration of an aircraft taking off.

This is a 30 or more second process, during which you can feel the force pushing you back into your seat and also you feel that acceleration is an integral part of this process. To me this is more intuitive than a short period of acceleration that is indistinguishable from velocity.

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    $\begingroup$ That is true. In the punching scenario, velocity is almost indistinguishable from acceleration. However, imagining constant velocity in a plane vs acceleration, and with the foundational ideas of relativity where motion with constant velocity is indistinguishable with a stationary frame of reference starts to make sense. Thank you. $\endgroup$ – StopReadingThisUsername Mar 20 '17 at 11:40
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If you apply a force of $1N$ of a body of $1kg$, it will move with $1\frac{m}{s^2}$ acceleration. Alternatively, if a body of 1kg is moving with acceleration of $1\frac{m}{s^2}$ then the force acting on the body is $1N$.

But if a body has a constant speed then there is not net force acting on it. A force can bring acceleration to a body motion, and acceleration means change in velocity.

According to newtons second law of motion,

$F =\frac{dP}{dt}$

$F=m\frac{dv}{dt}+v\frac{dm}{dt}$

Considering, $\frac{dm}{dt}=0$, then

$F=ma$

When you punch someone, your fist has some certain momentum at the time of impact. When you fist lands on your opponent, according to Newton's third law of motion, your fist apply force on the opponent and the opponent apply equal force back at your fist.

Its all about the impulse, $I=F\Delta{T}=\Delta{P}$

If you increase the time of impact, then the force will decrease. For a short amount of time, impulse can be enormously large in magnitude.

That is also why the fielders ,in cricket, swing their hands in the direction of the ball's trajectory so that they could decrease the impulse. If a fielder doesn't move his hands and tries to catch the ball with still hands, he might get hurt pretty bad, as in this case the impulse in high due to short time interval of impact.

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Assume you had been launched into space by a cannon, and your velocity was 10 m/s. When you're in space, you don't slow down because there are no forces such as gravity or drag acting on you, so you continue moving at 10 m/s forever. Does this mean a force is constantly acting on you?

If there were no resistance forces, an applied force would not be needed to maintain velocity.

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