Unless indicated otherwise, assume the speed of sound in air to be v = 344 m/s. The shock-wave cone created by a space shuttle at one instant during its reentry into the atmosphere makes an angle of 58.0o with its direction of motion. The speed of sound at this altitude is 331 m/s. (a) What is the Mach number of the shuttle at this instant, and (b) how fast (in m/s and in mi/h) is it traveling relative to the atmosphere? (c) What would be its Mach number and the angle of its shock-wave cone if it flew at the same speed but at low altitude where the speed of sound is 344 m/s?

Solution 56E Step 1: a) Since it is making an angle with the cone, we can write, sin = v/v spaceship The equation for the Mach number, M = u/v u - local flow velocity v - speed of sound But, here, u = v spaceship v = 331 m/s at highest altitude Therefore, we can write, v spaceshipv / sin from the first equation. That is, v spaceship31 m/s / sin 58 = 331 m/s / 0.8480 vspaceship90.33 m/s Therefore, Mach number, M = 390.33 m/s / 331 m/s = 1.179 Step 2: b) The speed of the spaceship is = 390.33 m/s We know that, 1 s = 1/3600 hr 1 m = 1/1000 km Therefore, 1 m/s = (1/1000) km / (1/3600) hr = 18 /5 km/hr Substituting this and converting the units we get, vspaceship 90.33 × (18/5) km/hr = 1405.19 km/hr km = 1.61 miles Therefore, v spaceship(1405.19 /1.61) miles / hour = 872.79 mph