(The parameter
represents how rapidly the force decreases to zero.) The average force is
where
. Since we already have a numeric value for
, we can use the result of the integral to obtain
.
Choosing
(this is a common choice, as you will see in later chapters), and guessing that
, this integral evaluates to
Thus, the maximum force has a magnitude of
The complete force function, including the direction, is
This is the force Earth applied to the meteor; by Newton’s third law, the force the meteor applied to Earth is
which is the answer to the original question.
Significance
The graph of this function contains important information. Let’s graph (the magnitude of) both this function and the average force together (
[link] ).
A graph of the average force (in red) and the force as a function of time (blue) of the meteor impact. The areas under the curves are equal to each other, and are numerically equal to the applied impulse.
Notice that the area under each plot has been filled in. For the plot of the (constant) force
, the area is a rectangle, corresponding to
. As for the plot of
F (
t ), recall from calculus that the area under the plot of a function is numerically equal to the integral of that function, over the specified interval; so here, that is
. Thus, the areas are equal, and both represent the impulse that the meteor applied to Earth during the two-second impact. The average force on Earth sounds like a huge force, and it is. Nevertheless, Earth barely noticed it. The acceleration Earth obtained was just
which is completely immeasurable. That said, the impact created seismic waves that nowadays could be detected by modern monitoring equipment.
The benefits of impulse
A car traveling at 27 m/s collides with a building. The collision with the building causes the car to come to a stop in approximately 1 second. The driver, who weighs 860 N, is protected by a combination of a variable-tension
seatbelt and an
airbag (
[link] ). (In effect, the driver collides with the seatbelt and airbag and
not with the building.) The airbag and seatbelt slow his velocity, such that he comes to a stop in approximately 2.5 s.
What average force does the driver experience during the collision?
Without the seatbelt and airbag, his collision time (with the steering wheel) would have been approximately 0.20 s. What force would he experience in this case?
The motion of a car and its driver at the instant before and the instant after colliding with the wall. The restrained driver experiences a large backward force from the seatbelt and airbag, which causes his velocity to decrease to zero. (The forward force from the seatback is much smaller than the backward force, so we neglect it in the solution.)
Strategy
We are given the driver’s weight, his initial and final velocities, and the time of collision; we are asked to calculate a force. Impulse seems the right way to tackle this; we can combine
[link] and
[link] .