How does the acceleration of a cart depend on the total mass if the net force is constant?
The answer above is correct: if the force is constant, the mass is inversely proportional to the acceleration. So if the mass increases, the acceleration decreases, and vice versa.
This comes from the second Newton's Law, that states that the net force equals mass times acceleration:
`F = m_1*a_1` , where `m_1` is the original mass and `a_1` is the original acceleration.
If the mass and acceleration change, but the force remains the same, then
`F = m_2*a_2` , where `m_2` is the new mass and `a_2` is the new acceleration.
Combining the two equations, we get
`m_1*a_1 = m_2*a_2`
If we divide this equation by `m_1*a_2` , it becomes
`a_1/a_2 = m_2/m_1` ,
which means that the acceleration is inversely proportional to mass.
This is why mass is also called "the measure of inertia". Inertia is the tendency of an object to resist the change in its motion, or the change in how fast it moves. The objects with larger mass have larger inertia, so they are harder to accelerate.
This is Newton's second law of motion in terms of the change in momentum:
`(P_f -P_i)/t = F`net
By using knowledge that momentum is mass times velocity, this equation can be manipulated into this form:
`(V_f - V_i)/t = F/m`
Again, by using knowledge that change in velocity over time is acceleration, we get that:
`a = F/m`(Remember that the force is still net)
From this we can see that m is inversely proportional to a since Fnet is constant. Increasing mass causes a decrease in the acceleration.