Question

object x and object y are interacting with each other. both objects are also interacting with other objects. based on this information, which quantities must be equal in magnitude at all times? choose all answers that apply: a the force (f) from x on y and the force from y on x b the net force (σf) on x and the net force on y c the acceleration (a) of x and the acceleration of y d the velocity (v) of x and the velocity of y

Explanation:

Step1: Analyze Newton's Third Law

Newton's third law states that for every action, there is an equal and opposite reaction. So, the force from X on Y and the force from Y on X are an action - reaction pair. According to the law, \(F_{X
ightarrow Y}=-F_{Y
ightarrow X}\), which means their magnitudes \(|F_{X
ightarrow Y}| = |F_{Y
ightarrow X}|\) are equal at all times.

Step2: Analyze net force

The net force on an object \(\sum\vec{F}=m\vec{a}\). Since the masses of \(X\) and \(Y\) are not given to be equal and they are also interacting with other objects (so the other - object - induced forces on \(X\) and \(Y\) can be different), \(\sum\vec{F}_X\) and \(\sum\vec{F}_Y\) are not necessarily equal.

Step3: Analyze acceleration

From \(\sum\vec{F}=m\vec{a}\), even if \(\sum\vec{F}_X=\sum\vec{F}_Y\) (which is not the case here as shown above), if \(m_X
eq m_Y\), then \(\vec{a}_X=\frac{\sum\vec{F}_X}{m_X}\) and \(\vec{a}_Y=\frac{\sum\vec{F}_Y}{m_Y}\) will be different. Also, since \(\sum\vec{F}_X
eq\sum\vec{F}_Y\) in general, \(a_X\) and \(a_Y\) are not equal.

Step4: Analyze velocity

Velocity \(\vec{v}=\vec{v}_0+\vec{a}t\). Since \(\vec{a}_X
eq\vec{a}_Y\) (from the acceleration analysis) and initial velocities \(\vec{v}_{0X}\) and \(\vec{v}_{0Y}\) are not given to be equal, \(\vec{v}_X\) and \(\vec{v}_Y\) are not equal.

Answer:

A. the force (\(F\)) from \(X\) on \(Y\) and the force from \(Y\) on \(X\)