How exactly does mass affect speed?

Question

I konw that mass affects weight (force), so how does that relate to speed? F=ma. so how does all this affect speed?

Answer 1

Mass doesn't affect speed directly. It determines how quickly an object can change speed (accelerate) under the action of a given force. Lighter objects need less time to change speed by a given amount under a given force.

Alternatively, mass determines how strong a force has to be to accelerate an object at a given rate. Lighter objects can do with weaker force to change speed by a given amount in a given amount of time.

Thus, mass is a measure of object's inertia which is resistance to changes in object's motion. This is what the equation

\begin{equation} F=ma \end{equation}

means.

Note that speed is relative (i.e. depends on the choice of the frame of reference) and, in the framework of Newtonian mechanics, for each object there is a frame of reference in which it is moving at arbitrarily high velocity.

Answer 2

Let us first assume that a constant force, $F$, is applied to a stationary body of mass $m$ from $t = 0$ to $t = T$.

Let us begin with the equation $F = ma$.

This can be re-written as $F = m\frac{dv}{dt}$.

After re-arranging the equation, we arrive at:

$\frac{dv}{dt} = F/m$. we can then integrate both sides with respect to time to give:

$v = \int_0^T \frac{F}{m}\mathrm dt$, where the limits of integration apply over the time the force is applied.

Since we have assumed that $F$ is constant between $t=0$ and $t=T$, we can calculate the integral to arrive at:

$v = \frac{F}{m}T$, assuming the mass is initially at rest before the force is applied.

As we can see, if the mass was doubled, the final velocity achieved at $t=T$ will be halved.

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Secondly, one may consider how mass effects speed by considering kinetic energy. The formula for kinetic energy is given by:

$KE = \frac{1}{2}mv^2$.

Therefore, if a photon losses all its energy, $hf$, to a free particle of mass, $m$, the particle will travel at a speed of $\sqrt{2\frac{hf}{m}}$ after gaining the energy from the photon. If however a photon loses the same amount of energy to a particle of double the mass, $2m$, the particle will travel at a velocity of $\sqrt{\frac{hf}{m}}$, which is slower by a factor of $\sqrt{2}$.

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In conclusion, we have seen that the final velocity of a heavy particle after the application of a constant force will be lower than that of a light particle. Similarly, a heavy particle will travel at a lower velocity than a light particle, given an equal amount of energy.