4.2. Products and Quotients

Ever wondered what the speed of a bullet is? Let's calculate it! The picture below is an actual photo of a rifle bullet in flight. A flash was used twice with a time interval of 1 millisecond. The white dot on the left is the bullet at the time of the first flash. The dot on the right is the same bullet 1.00 ms ± 0.03 ms later, at the time of the second flash.

Bullet flying over a ruler. Permission granted from fotoopa.

First, let's determine the distance traveled by the bullet. The position of the bullet on the left is at 23.0 cm ± 0.5 cm. The position of the bullet on the right is 37.5 cm ± 0.5 cm. The distance traveled D is then 14.5 cm. Since we subtracted two quantities to obtain the distance, we should add the errors. So our error on distance is 1.0 cm and our result for D is:

As you already know, the second expression is the result written with the relative error, which in this case is about 7%. The time of flight T between the two points is equal to the time interval between the two flashes, which is known to be 1 millisecond with the relative error of 3%.

The velocity V is distance over time.

The central value for the velocity is then 14.5 cm/msec (or remembering that 1 meter is equal to 100 cm, and 1 second is equal to 1000 milliseconds), V=145 m/s.¹ That was easy. But how precise is our answer? It is clear that our final error on V should be somehow larger then the individual errors on D and T since we combine the two to get V. However, we cannot just add our absolute errors as we did in the previous section since the errors have different units. The relative errors have no units; can we add them? Indeed, we can. To see that, consider the largest possible value for the velocity V:

You might remember the following formula from your mathematics course

The above formula is true for a small compared to 1. We use this formula for our calculation of the largest velocity. In our case, a = 0.03. We see that 1 / (1-0.03) = 1.0309 is in fact very close to 1.03, and we can write:

After multiplying out the parentheses, we obtain

We can safely discard the last number 0.0021 since it is much smaller than the rest. Our answer for the largest velocity is then

An almost identical calculation for the lowest velocity ( try to do it yourself! ) gives

Finally, we can quote our final answer with the correct error in its full glory

Converting from relative to absolute errors

Now we know how to calculate the error on the quotient of two quantities. We should add the relative, and not the absolute, errors of the quantities. What about a product of two quantities? The good news is that the rule is the same for products as for quotients. We add the relative errors. We won't go through the derivation of the rule since it's really almost entirely identical to the one we gave for the quotients. Just remember that in the case of products and quotients, you always add the relative errors.

¹ 145 m/s is very slow for a bullet. Typical speeds are > 300 m/s.