# Why is acceleration affected by mass?

Jul 9, 2014

The equation
$F = m a$
describes the relationship between force, mass, and acceleration. Your question seems to be asking how the mass can change the acceleration of an object. But that is not really what the equation is telling us. Many textbooks like to introduce the concept this way:
$a = \frac{F}{m}$
It is exactly the same formula with a simple algebraic change. In this form, the cause and effect relationship is more clearly seen. Read it this way: The acceleration (a) will result when a force (F) is exerted on an object with mass (m). In this form, the equation has the two things you can control (force and mass) on one side, and the thing you observe as a result (acceleration) on the other side.

If I want something to accelerate I can exert a force on it. The magnitude of the acceleration will depend on how much force I use, and how heavy the object is. If I want to accelerate it twice as much, I will need to use twice as much force.

Example 1:
If two friends are pushing identical chunks of ice across a frozen lake, the acceleration of each chunk will be the same only if the force on each chunk is the same.

Example 2:
If two friends are pushing chunks if ice across a frozen lake and one chunk weighs twice as much as the other chunk, then the person pushing the more massive chunk will need to exert twice as much force to keep up with her friend.

From Newton's Laws, the acceleration can be calculated:
${a}_{1} = {F}_{1} {m}_{1}$
${a}_{2} = {F}_{2} {m}_{2}$

We know the relative mass of the two objects:
${m}_{2} = 2 {m}_{1}$

If we want the same acceleration for each object then we know that: ${a}_{1} = {a}_{2}$

After a little figuring we can show that ${F}_{2} = 2 {F}_{1}$
Said again: If you want to accelerate something that is twice as heavy, you will need twice as much force.