From Newton's first law, and by analysing the inertia of an object, what must we do to accelerate an object?

Apply a force!

What if the object is really, really massive?

Apply a really, really large force?

Let's see if we can develop an equation for it.

If we look at the table below, we see a number of cars, and their mass, 0-100 km/h time and power output.

Think About It:

What kind of conclusions can you draw from the information in the table?

If we look at the table there are two ways we can draw some relationships about the F_net, mass, and acceleration.

If we compare the BMW and the Lamborghini, we can see that their masses are comparable, but the force they can apply are significantly different, and not surprisingly, the Lamborghini is quicker than the BMW.

So, we can draw a conclusion, that acceleration is proportional to the F_net.

a = k F_net

_{(we use k in here to represent a bunch of different things that we do not know at this point, k is called a proportionality constant)}

We can also compare the BMW and the Hayabusa.  Their power is similar, but again, the performance is much different.  So now the acceleration depends on how little mass we have.

The less massive something is, the quicker that object can accelerate (so long as the force is constant).  So we can say that the acceleration is inversely proportional to the mass.

a = k/m

We can then pull all of this together.  If we have two different proportionality relations, we can move them into one big equation, where,

a = k F_net/m

But, since our units work out perfectly (Force is in N = kg-m/s², acceleration is in m/s², and mass is in kg), k just equals 1, and drops out of our equation.  So,

a = F_net/m

We can then rearrange this formula to solve for F_net.

F_net = m a

Which gives us the equation for Newton's Second Law:

The acceleration of an object as produced by a net force is directly
proportional to the magnitude of the net force, in the same direction
as the net force, and inversely proportional to the mass of the object.

Now, one question we need to ask ourselves, is:

Does Newton's First Law agree with Newton's Second Law?

The answer is yes.  First Law states that if there is no force acting on an object, the acceleration of that object will be zero.  If we plug "zero" into acceleration:

F_net = m a
F_net = M (0)
F_net = 0

So, we can see right there, that if there is zero acceleration, the net force on the object has to be zero.  Therefore, Newton's First Law DOES agree with second law!