Newton’s third law is one quantitative relationship you can use to solve problems of interacting objects. In addition, we frequently have other information about the motion in a problem. For example, if two objects A and B move together, their accelerations are constrained to be equal: $\vec{a}{}_{\text{A}}=\vec{a}{}_{\text{B}}$. A well-defined relationship between the accelerations of two or more objects is called an acceleration constraint. It is an independent piece of information that can help solve a problem.

In practice, we’ll express acceleration constraints in terms of the x- and y-components of $\vec{a}$ Consider the car being towed in FIGURE 7.14. This is one-dimensional motion, so we can write the acceleration constraint as

Because the accelerations of both objects are equal, we can drop the subscripts C and T and call both of them $a_x$.

FIGURE 7.14 The car and the truck have the same acceleration.

Don’t assume the accelerations of A and B will always have the same sign. Consider blocks A and B in FIGURE 7.15. The blocks are connected by a string, so they are constrained to move together and their accelerations have equal magnitudes. But A has a positive acceleration (to the right) in the x-direction while B has a negative acceleration (downward) in the y-direction. Thus the acceleration constraint is

This relationship does not say that $a_{\text{A}x}$ is a negative number. It is simply a relational statement, saying that $a_{\text{A}x}$ is $(-1)$ times whatever $a_{\text{B}y}$ happens to be. The acceleration $a_{\text{B}y}$ in Figure 7.15 is a negative number, so $a_{\text{A}x}$ is positive. In some problems, the signs of $a_{\text{A}x}$ and $a_{\text{B}y}$ may not be known until the problem is solved, but the relationship is known from the beginning.

FIGURE 7.15 The string constrains the two objects to accelerate together.