Video Transcript
An object moves along a straight line at a speed of two plus or minus 0.1 meters per second for 20 plus or minus 0.5 seconds. Work out the distance that the object moves as well as the uncertainty in this value.
In this scenario then, we have some object moving along a straight line. And it moves at a speed we’ll call 𝑣 for an amount of time we’ll call 𝑡. We want to figure out the distance that the object travels as well as the uncertainty in that distance. We can recall that in general the average speed of an object equals the distance that object travels divided by the time taken to travel that distance. Multiplying both sides of this equation by the time 𝑡 so that 𝑡 cancels out on the right, we find that the distance traveled by our object equals its average speed times the time taken to travel that distance.
For our scenario then, we can write that 𝑑 equals 𝑣 times 𝑡, which is equal to two plus or minus 0.1 meters per second times 20 plus or minus 0.5 seconds. The total distance traveled by our object will equal the product of two meters per second and 20 seconds. That equals 40 meters, but we know that this distance has some uncertainty attached to it. That’s because both the speed and the time that went into it have their own uncertainties.
We see that what we’re doing is taking two quantities, 𝑣 and 𝑡, that have their own uncertainties associated with them and multiplying them together to find a third quantity, in this case the distance 𝑑. In general, if we have two numbers 𝑎 and 𝑏 each with its own uncertainty, 𝜎 𝑎 and 𝜎 𝑏, respectively, and if we multiply these two numbers together to get a third number, we’ll call it 𝑐, then the uncertainty in that product, we’ll call it 𝜎 𝑐, equals 𝑏 times the uncertainty of 𝑎 plus 𝑎 times the uncertainty of 𝑏.
We can apply this rule to our particular situation. We can say that in a speed 𝑣, two is 𝑎 and 0.1 is 𝜎 𝑎, while in the time 𝑡, 20 we’ll call 𝑏 and 0.5 is the uncertainty in that value, 𝜎 sub 𝑏. The uncertainty in the product of 𝑣 and 𝑡 then is equal to 20, that’s 𝑏, times 0.1, that’s the uncertainty in 𝑎, plus two, that’s 𝑎, times 0.5, that’s the uncertainty in 𝑏. And then, when it comes to the units, we simply multiply these values together. Meters per second times seconds gives us final units of meters because the seconds in numerator and denominator cancel out. So then, 20 times 0.1 is equal to two, and two times 0.5 is equal to one. The overall uncertainty then is two plus one, or three meters.
We’ve therefore calculated both the distance that the object moves and the uncertainty in this value. The object moves a distance of 40 meters plus or minus three meters.
