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Chapter 12: Problem 58

The Vespertilionidae family of bats detect the distance to an object by timing how long it takes for an emitted signal to reflect off the object and return. Typically they emit sound pulses 3 ms long and 70 ms apart while cruising. (a) If an echo is heard 60 ms later $\left(v_{\text {sound }}=331 \mathrm{m} / \mathrm{s}\right),\( how far away is the object? (b) When an object is only \)30 \mathrm{cm}$ away, how long will it be before the echo is heard? (c) Will the bat be able to detect this echo?

Short Answer

Expert verified
Answer: (c) The bat will be able to detect the echo at 30 cm away if the time it takes for the echo to be heard is less than the time between pulses (70 ms).

Step by step solution

01

(a) Finding the distance to the object

: To determine the distance of the object from the bat when it hears the echo after 60 ms (0.06 s), we can use the speed of sound (\(v_{sound} = 331 m/s\)) and the time it takes to hear the echo following the formula: total distance = speed 脳 time. Since the sound needs to travel to the object and back, we will divide the total distance by 2 to get the distance to the object. The total distance traveled by the sound is: Total distance = \(v_{sound}\) 脳 time Total distance = \(331 m/s\) 脳 \(0.06 s\) Distance to the object = Total distance / 2
02

(b) Finding the time before the echo is heard

: We are given that the object is 30 cm (0.3 m) away from the bat. To find the time it takes for the echo to be heard by the bat, we can once again use the relationship between speed, distance, and time: time = distance / speed. Since the sound travels to the object and back, we need to multiply the distance by 2 to get the total distance: Total distance = distance to object 脳 2 Total distance = \(0.3 m\) 脳 2 Time before the echo is heard = Total distance / \(v_{sound}\)
03

(c) Determining if the bat can detect the echo

: Finally, we need to determine if the bat can detect the echo when the object is 30 cm away. From part (b), we found the time it takes for the echo to be heard at this distance. We have the given information that the sound pulses are 3 ms long and 70 ms apart, and we will use this to determine if the bat can detect the echo. If the time before the echo is heard is less than the time it takes between pulses (70 ms), then the bat will be able to detect the echo. Otherwise, the bat won't be able to detect the echo. Comparing the time between pulses and the time before the echo is heard (from part (b)), we can determine if the bat can detect the echo.

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