Question Video: Finding the Uncertainty in an Object’s Kinetic Energy Physics • First Year of Secondary School

Video Transcript

An object has a mass of two plus or minus 0.1 kilograms and is moving at a speed of three plus or minus 0.1 meters per second. What is the uncertainty in the kinetic energy of the object? Start by calculating the uncertainty in 𝑣 squared and then calculate the uncertainty in one-half 𝑚 times 𝑣 squared.

In this example, we have an object with a mass we’ll call 𝑚 moving along at a speed we’ll call 𝑣. Both the mass and the speed of our object are given with some amount of uncertainty. We want to solve for the uncertainty in the object’s kinetic energy, which we can recall is equal to one-half its mass times its speed squared. Our problem statement tells us to begin by calculating the uncertainty in 𝑣 squared.

Since the speed 𝑣 of our object is three plus or minus 0.1 meters per second, that means we can write 𝑣 squared this way. We see that to calculate 𝑣 squared, we’ll be multiplying this quantity by this quantity and that both of these quantities have an uncertainty associated with them. We may be able to tell that the value of 𝑣 squared itself will be three times three, or nine. But what about combining the uncertainties in these two values to give an overall uncertainty?

To better understand how to do this, let’s imagine that we have two quantities, each of which has some associated uncertainty. There’s the quantity 𝑎 plus or minus the uncertainty in 𝑎 and the quantity 𝑏 plus or minus the uncertainty in 𝑏. If we decide to multiply 𝑎 and 𝑏 together to create some other quantity 𝑐, then just as 𝑎 and 𝑏 have associated uncertainties, so will their product 𝑐.

The uncertainty in the quantity 𝑐, we’ll call it 𝜎 sub 𝑐, is equal to 𝑏 times the uncertainty of 𝑎 plus 𝑎 times the uncertainty of 𝑏. Mathematically, this is equivalent to saying that the uncertainty in 𝑐 divided by 𝑐 equals the uncertainty in 𝑎 divided by 𝑎 plus the uncertainty in 𝑏 divided by 𝑏.

When it comes then to solving for the uncertainty in 𝑣 squared, we can think of this three as quantity 𝑎 and this uncertainty of 0.1 as 𝜎 sub 𝑎. And then in the next factor, we can think of this three as 𝑏 and this uncertainty of 0.1 as 𝜎 sub 𝑏. So, 𝑣 squared is equal to the quantity 𝑎 times 𝑏, or three times three, that’s nine, plus or minus an uncertainty. According to our equation, that uncertainty equals 𝑏 or three times the uncertainty in 𝑎, 0.1, plus 𝑎, which is three, times the uncertainty in 𝑏, also 0.1.

As a side note, this equation we’re using to calculate uncertainty lets us correctly calculate the numerical value of that uncertainty, but it doesn’t work for the units involved. We’ll need to keep track of those ourselves. We know the units for 𝑣 squared will be meters per second quantity squared. Looking inside of our square braces, we have two instances of three being multiplied by 0.1. This gives 0.3 in each case, which when added together equals 0.6. We can say then that the speed of our object squared is equal to nine plus or minus 0.6 meters squared per second squared.

As we can see though, that’s just one part of our object’s kinetic energy. If we take 𝑣 squared and we multiply it by our object’s mass, then, including uncertainties, that’s equal to two plus or minus 0.1 kilograms multiplied by nine plus or minus 0.6 meters squared per second squared. Once again, we can assign these values of 𝑎, 𝜎 sub 𝑎 and 𝑏 and 𝜎 sub 𝑏. We know that 𝑚 times 𝑣 squared will be equal to two times nine, or 18, plus or minus the uncertainty of nine times 0.1, that’s 𝑏 times the uncertainty in 𝑎, plus two times 0.6, that’s 𝑎 times the uncertainty in 𝑏. And once again, the units need to fend for themselves. Those units are kilograms meter squared per second squared.

We know that nine times 0.1 is 0.9 and two times 0.6 is 1.2. So, the total uncertainty in this quantity is 2.1. Even though we now know this, we’re not quite at our final answer. First, because we can simplify these units, a kilogram times a meter squared per second squared is equal to one joule of energy. But then, more importantly, we haven’t quite yet calculated the full kinetic energy of our object. Note that so far we know 𝑚 times 𝑣 squared, but that kinetic energy is one-half 𝑚 times 𝑣 squared.

It’s at this point that we’ll want to be very careful. Once again, we’re multiplying two numbers: one-half and 𝑚 times 𝑣 squared. If we enter in these values including their uncertainties, we would write one-half plus or minus zero because one-half is an exact number with no uncertainty. And then for 𝑚 times 𝑣 squared, we have 18 plus or minus 2.1 joules. Once again then, we’re multiplying two numbers that have uncertainties, even though the uncertainty of one is zero.

The kinetic energy of this object then is one-half times 18, or nine, plus or minus 𝑏 times the uncertainty in 𝑎, that’s 18 times zero, plus 𝑎 times the uncertainty in 𝑏, that’s one-half times 2.1. We can see right away that 18 times zero is zero. And so that term doesn’t contribute anything to our overall uncertainty. One-half times 2.1 then, which is 0.15, is our overall uncertainty. Our final answer will include the units because this uncertainty is in those units.

The uncertainty in the kinetic energy of the given object is 1.05 joules.