Civil engineers use a transit theodolite when surveying. One version of this dev

Question

Civil engineers use a transit theodolite when surveying. One version of this device determines distance by measuring the time required for an ultrasonic pulse to reach a target, reflect from it, and return. When calibrated properly, the device uses the speed of sound appropriate for the ambient air temperature to determine the distance between the theodolite and the target. Suppose a theodolite is calibrated so that it gives the correct value for a distance when used at a temperature of 284 K. When used at a temperature of 292 K, however, the device gives an incorrect value for the same distance. What percentage error would there be in the incorrect value measured by the theodolite? Is the incorrect value smaller than or greater than the correct value? Assume that air behaves as an ideal gas, so that

Explanation / Answer

keeping all the parameters constant, v is directly proportional to sqrt(T).

let speed at T=284 K be v1 and speed at T=292 K be v2.

then v1/v2=sqrt(284/292)=0.98621

device is calibrated for 284 K i.e. for the speed of ultrasnoic pulse being v1.

but when temperature changes to 292 K, the speed of the pulse becomes v2 although the device measures the distance still based on v1.

now , as the actual pulse has a speed v2 which is greater than v1, time taken will be less.

and as calculations are done as per v1, distance measured=v1*time taken will be lesser than the actual distance.

let distance be d.

then time taken=d/v2

measured distance=v1*(d/v2)

then percentage error=(d-(d*(v1/v2))*100/d

=(1-0.98621)*100/1=1.379%

the incorrect value is smaller than the correct value.

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