9.2 Impulse and collisions  (Page 3/10)

(The parameter $\tau$ represents how rapidly the force decreases to zero.) The average force is

where $\text{Δ}t=t_{\text{max}}-0\text{s}$ . Since we already have a numeric value for $F_{\text{ave}}$ , we can use the result of the integral to obtain $F_{\text{max}}$ .

Choosing $\tau =\frac{1}{e}{t}_{\text{max}}$ (this is a common choice, as you will see in later chapters), and guessing that $t_{\text{max}}=2\text{s}$ , 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

Notice that the area under each plot has been filled in. For the plot of the (constant) force $F_{\text{ave}}$ , the area is a rectangle, corresponding to $F_{\text{ave}}\text{Δ}t=J$ . 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 ${\displaystyle {\int }_0^{t_{\text{max}}}F}(t)dt=J$ . 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

1. What average force does the driver experience during the collision?
2. 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?

Strategy