If a car stops suddenly, you may be “thrown” into the windshield if you’re not wearing your seat belt. You have a very real forward acceleration relative to the car, but is there a force pushing you forward? A force is a push or a pull caused by an identifiable agent in contact with the object. Although you seem to be pushed forward, there’s no agent to do the pushing.

The difficulty—an acceleration without an apparent force—comes from using an inappropriate reference frame. Your acceleration measured in a reference frame attached to the car is not the same as your acceleration measured in a reference frame attached to the ground. Newton’s second law says $\vec{F}{}_{\text{net}}=m\vec{a}$. But which $\vec{a}?$ Measured in which reference frame?

We define an inertial reference frame as a reference frame in which Newton’s first law is valid. If $\vec{a}=\vec{0}$ (an object is at rest or moving with constant velocity) only when $\vec{F}{}_{\text{net}}=\vec{0}$, then the reference frame in which $\vec{a}$ is measured is an inertial reference frame.

Not all reference frames are inertial reference frames. FIGURE 5.20a shows a physics student cruising at constant velocity in an airplane. If the student places a ball on the floor, it stays there. There are no horizontal forces, and the ball remains at rest relative to the airplane. That is, $\vec{a}=\vec{0}$ in the airplane’s reference frame when $\vec{F}{}_{\text{net}}=\vec{0}$. Newton’s first law is satisfied, so this airplane is an inertial reference frame.

FIGURE 5.20 Reference frames.

The physics student in FIGURE 5.20b conducts the same experiment during takeoff. He carefully places the ball on the floor just as the airplane starts to accelerate down the runway. You can imagine what happens. The ball rolls to the back of the plane as the passengers are being pressed back into their seats. Nothing exerts a horizontal contact force on the ball, yet the ball accelerates in the plane’s reference frame. This violates Newton’s first law, so the plane is not an inertial reference frame during takeoff.

In the first example, the plane is traveling with constant velocity. In the second, the plane is accelerating. Accelerating reference frames are not inertial reference frames. Consequently, Newton’s laws are not valid in an accelerating reference frame.

The earth is not exactly an inertial reference frame because the earth rotates on its axis and orbits the sun. However, the earth’s acceleration is so small that violations of Newton’s laws can be measured only in very careful experiments. We will treat the earth and laboratories attached to the earth as inertial reference frames, an approximation that is exceedingly well justified.

To understand the motion of the passengers in a braking car, you need to measure velocities and accelerations relative to the ground. From the perspective of an observer on the ground, the body of a passenger in a braking car tries to continue moving forward with constant velocity, exactly as we would expect on the basis of Newton’s first law, while his immediate surroundings are decelerating. The passenger is not “thrown” into the windshield. Instead, the windshield runs into the passenger!