The Accelerations Produced by Action and Reaction Forces

Example 4 The Accelerations Produced by Action and Reaction Forces

Suppose that the mass of the spacecraft in Figure 4.7 is m_S = 11 000 kg and that the mass of the astronaut is m_A = 92 kg. In addition, assume that the astronaut exerts a force of

on the spacecraft. Find the accelerations of the spacecraft and the astronaut.

Reasoning According to Newton's third law, when the astronaut applies the force

to the spacecraft, the spacecraft applies a reaction force

to the astronaut. As a result, the spacecraft and the astronaut accelerate in opposite directions. Although the action and reaction forces have the same magnitude, they do not create accelerations of the same magnitude, because the spacecraft and the astronaut have different masses. According to Newton's second law, the astronaut, having a much smaller mass, will experience a much larger acceleration. In applying the second law, we note that the net force acting on the spacecraft is

, while the net force acting on the astronaut is

Solution

Problem-solving insight

Even though the magnitudes of the action and reaction forces are always equal, these forces do not necessarily produce accelerations that have equal magnitudes, since each force acts on a different object that may have a different mass.

Using the second law, we find that the acceleration of the spacecraft is

The acceleration of the astronaut is