Question

A meteorite hits the surface of the ocean at a speed of \(8.80 \mathrm{~km} / \mathrm{s}\). What is the angle of the shock wave it produces (a) in the air just before hitting the ocean surface and (b) in the ocean just after entering? Assume that the speeds of sound in air and in water are \(343 \mathrm{~m} / \mathrm{s}\) and \(1560 \mathrm{~m} / \mathrm{s},\) respectively.

Short answer

(a) Approximately 2.24 degrees and (b) approximately 10.18 degrees.

Step-by-step solution

Convert the meteorite speed to meters per second

Since the given meteorite speed is in kilometers per second, we first need to convert it to meters per second to be consistent with the speed of sound values provided: \(8.80\ \mathrm{km}/\mathrm{s} \cdot \frac{1000\ \mathrm{m}}{1\ \mathrm{km}} = 8800\ \mathrm{m}/\mathrm{s}\) Now, we have the meteorite speed in meters per second, which is 8800 m/s.

Calculate the Mach angle in air

Now, we can use the Mach Angle formula and plug in the speed of sound in air as well as the speed of the meteorite. \(sin(\theta_{air}) = \frac{343\ \mathrm{m}/\mathrm{s}}{8800\ \mathrm{m}/\mathrm{s}}\) Solve for the Mach angle in air: \(\theta_{air} = sin^{-1}\left(\frac{343\ \mathrm{m}/\mathrm{s}}{8800\ \mathrm{m}/\mathrm{s}}\right)\) \(\theta_{air} \approx 2.24^\circ\) The angle of the shock wave in air just before hitting the ocean surface is approximately 2.24 degrees.

Calculate the Mach angle in water

Next, we can use the Mach Angle formula and plug in the speed of sound in water as well as the speed of the meteorite. \(sin(\theta_{water}) = \frac{1560\ \mathrm{m}/\mathrm{s}}{8800\ \mathrm{m}/\mathrm{s}}\) Solve for the Mach angle in water: \(\theta_{water} = sin^{-1}\left(\frac{1560\ \mathrm{m}/\mathrm{s}}{8800\ \mathrm{m}/\mathrm{s}}\right)\) \(\theta_{water} \approx 10.18^\circ\) The angle of the shock wave in water just after entering the ocean is approximately 10.18 degrees. In conclusion, the angle of the shock wave produced by the meteorite hitting the ocean surface is (a) approximately 2.24 degrees in the air just before hitting the ocean surface and (b) approximately 10.18 degrees in the ocean just after entering.