# Why is force dependent on acceleration, not velocity?

According to $F = ma$, force is a result of acceleration and mass, right?

However, I don't understand why velocity is not used instead of acceleration. A train moving at 100 miles/hour will still impart a great force on you even though it has no acceleration. Further, dropping a book at 10ft will impart a greater force on the gorund than dropping it at 1ft. So it seems that velocity would influence the force more than the acceleration would.

Why is this not the case?

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force is a result of acceleration and mass, right? Wrong. The net force equals the time rate of change of momentum: $\vec F_{net} = \frac{d\vec p}{dt} = m\vec a$. Based on the way your question is phrased, I wonder if you're conflating the notion of impulse with force: en.wikipedia.org/wiki/Impulse_%28physics%29 – Alfred Centauri Mar 18 '14 at 2:27
The only way to get force from an impact velocity is F = mv^2/d. Which works great, but only if you can figure out what the d is. No ideas? It is how long it takes for the two masses to exchange energy and momentum. Which is really hard to figure out ahead of time, and one good reason that this approach isn't used much. – RBarryYoung Mar 18 '14 at 2:40

## 1 Answer

The $F$ in the equation $F=ma$ is not the force that would be exerted by the object if it were to hit something else. Instead, $F$ represents the net force acting on the object that must be present in order to produce the current acceleration $a$ of the object. A better way to write Newton's second law is $$F_\text{net}=ma,$$ since it shows explicitly which force is being represented on LHS of the equation is.

In your train example, if the train is traveling at a constant velocity of 100 mph, the acceleration is zero, and by Newton's second law the net force is also zero. But this has no bearing on what force the train would exert on something if it collided.

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How would you then calculate the force that an object would exert if it collided on something? – 1110101001 Mar 18 '14 at 2:35
And wouldn't it be better to rewrite newtons second law as a = F/m? That way the interpretation that acceleration is proportional to the force applied and inversely proportional to the mass is much clearer? – 1110101001 Mar 18 '14 at 2:36
@user2612743 Some educators prefer to write it that way; they say it makes the case that acceleration is "caused by" net force a little clearer. So, yes, to some. – BMS Mar 18 '14 at 2:58
@user2612743 To calculate the force in collisions you us the impulse divided by time, or $F_{net}=\Delta P/\Delta t$. – Ruben Mar 19 '14 at 6:20