# Does Inertia really define the resistant ability of a body

I was reading the chapter Newton's 1st law of Motion , where I met the definition of Inertia.

The property of an Object by virtue of which it neither changes its state nor tends to change the state, is called Inertia.

What I made out of it, was That Inertia gives us the resisting ability of An object.

But doesn't this ability also depend on velocity. I know I'm referring to momentum but isn't inertia supposed to tell us which object has a greater resisting capability.

Hopefully, I'm clear enough.

Edit 1 : From Steeven's Answer I am certain that inertia doesn't depend on Velocity. But as far as I understand it, inertia is dependent on acceleration(For Force is directly proportional to inertia and Acceleration is directly proportional to Force).I am I correct with my observations ?

• Why do you think "this ability" depends on velocity? – ACuriousMind Aug 29 '16 at 12:46
• @ACuriousMind Because two objects of same mass, but with different velocity have different Resistance. Obviously, the one with greater velocity will have more resistance. – Imaginary Pumpkin Aug 29 '16 at 12:49
• That's not obvious to me at all, since velocity is relative. – ACuriousMind Aug 29 '16 at 12:52
• @ACuriousMind What do you mean by Velocity is relative – Imaginary Pumpkin Aug 29 '16 at 12:54

There are many kinds of "resistance". Resistance against heat conduction (thermal resistivity $\sigma$), resistance against light passing through (opacity $\alpha$), resistance against electric conduction (electrical resistivity $\rho$ or just resistance $R$) etc.

As you say, resistance against changes in motion is called inertia.

• In the case of linear (translational) motion, this inertia is called mass $m$, which resists (linear) acceleration $a$.
• In the case of rotational motion it is called moment of inertia $I$, which resists angular acceleration $\alpha$.

But note the words "changes in motion". Not just motion. The resistance against a change in velocity is not dependent on the velocity. Accelerating a resting object is just as hard as accelerating an already moving object.

• If a sattelite stands still with $v=0\;\mathrm{m/s}$, it is tough to accelerate it with $a=2\;\mathrm{m/s^2}$ so that it will move $v=4\;\mathrm{m/s}$ after 2 seconds.
• But another sattelite that passes by, which is already moving at $v=15\;\mathrm{m/s}$ is just as tough to accelerate with $a=2\;\mathrm{m/s^2}$, so that it will move $v=19\;\mathrm{m/s}$ after 2 seconds.

The "gain in velocity" so to speak is the point - that is, the change or the acceleration. Not the velocity itself.

All this is cleared out in Newton's 2nd law:

$$\sum F=ma$$

Acceleration needs force. No acceleration requires no force, so keeping up the current velocity is not hard. But changing it needs a force big enough to cause the needed acceleration $a$. And how hard that is depends on the inertia, in this case mass $m$ - larger mass causes a larger force to be needed to give a certain acceleration $a$.

Same goes for rotational motion:

$$\sum \tau =I\alpha$$

with $I$ being the inertia in this case (and $\tau$ the torques causing an angular acceleration $\alpha$)

• Just A Clarification on the last Line, How can the force required to accelerate(or decelerate) a still object be equal to the one required for a satellite. – Imaginary Pumpkin Aug 29 '16 at 12:58
• @Shashwat This is a good point. See my addition above. The point is, as Newton found out, that force is not connected to velocity in any way. Only to acceleration. Moving or not makes no difference. – Steeven Aug 29 '16 at 13:01
• A bullet is moving with a uniform velocity(a=0), then to stop it I need the exact force that is needed to stop a similar bullet moving with a comparatively lower velocity but with no acceleration? – Imaginary Pumpkin Aug 29 '16 at 13:06
• What does this acceleration refer to , the one of the object, or the one of the Force?(I guess that sums up the last question) – Imaginary Pumpkin Aug 29 '16 at 13:27
• To stop a bullet, you could let it hit a wall. It would have to stop in a very short time, which means a huge (negative) acceleration. That takes a large force (according to Newton's 2nd law). But you could also stop the bullet with a thick pillow or in a water bath (as is done in bullet-shell measurements). Then the bullet is slowed down over a much longer time, so the (negative) acceleration, which is the amount it slows down every second, is much smaller. This then requires a smaller force - but kept for a longer time of course. – Steeven Aug 29 '16 at 13:30

Equations of motion for linear accelerations.

a = acceleration, v = velocity, d = distance, t = time, 0 ==> initial condition

a = a0 = constant

v = v0 + a0*t

d = d0 + v0*t + (1/2)*a0*t^2

Symbol v0 in the above equations is the initial condition for the velocity. The acceleration is the rate of change for the velocity, and that rate of change of velocity is independent of v0, the initial condition.

F = force, m = mass

For a constant mass (i.e. not a rocket)

F = m0*a

a = F/m0

so acceleration is proportional to the applied force.

Velocity is not involved in the equation that relates Force to acceleration.

Going back to Inertia as resistance to change, in the absence of external forces, velocity is constant (i.e. the acceleration, or change in velocity during an elapsed time, is zero).

The greater the mass (inertia) the more force will be required to change the velocity of a given body. It doesn't really matter whether that body has an initial velocity that is fast, or is zero.

In your example, you were talking about changing the velocity over a large interval (fast to zero) or a small interval (slow to zero). It takes as much force (for a given amount of time) to slow something from 25 ft/s to 20 ft/s as it does to change the velocity from 5 ft/s to 0 ft/s. The difference is that the first will still be going 20 ft/s and the second will be at rest (relative to your inertial frame of reference).