Depth · one concept from first principles · step 2 of 4
A fires at a target m away and hears the bullet hit the target seconds after firing. B stands m from A and m from the target, and hears the bullet hit the target seconds after hearing A's gunshot. Find the speed of the bullet. (The bullet and the sound each travel at a constant speed.)
Let the bullet travel x m per second and sound y m per second. Every time in the story is a distance divided by one of these.
A time of travel is just distance over speed, so the unknown speeds appear in the denominators.
6.RP.A.3Analyze The UnitsThe bullet flies 1700 m to the target, then the sound of the hit travels the same 1700 m back to A.
A's waiting time splits into two legs: bullet out, sound back.
8.EE.C.8Draw A DiagramFrom the shot, the gunshot reaches B after 980/y seconds and the hit sound after 1700/x + 2000/y seconds, 5 seconds apart.
Put both sounds on one timeline starting at the shot, then take the difference.
8.EE.C.8Draw A DiagramBoth equations contain the same bullet time 1700/x. Subtracting the second from the first leaves only the speed of sound.
Treat each fraction as one block; the bullet block cancels exactly like a variable would.
8.EE.C.8Convert To AlgebraSubtracting B's equation from A's removes the bullet time and leaves .
Each fraction is a travel time, distance divided by speed, so is the same bullet flight in both equations.
Taking equal amounts away from equal amounts leaves equal amounts, so the left side of A's equation minus the left side of B's equals .
In that difference the bullet time is taken away from itself and vanishes, because subtraction undoes addition.
is pieces of size less of the same pieces, which is .
Sound takes 1700/340 = 5 seconds to come back to A, so the bullet took the other 2 seconds of A's 7.
Once one block is known, the other comes straight out of the first equation.
8.EE.C.8Convert To AlgebraWhen the unknowns sit in denominators, treat each whole fraction as the thing to eliminate -- the system is still linear in those blocks.
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