Question

Determine the ratio β = v/c for each of the following.

(a) A car traveling 120 km/h.
(b) A commercial jet airliner traveling 270 m/s.
(c) A supersonic airplane traveling mach 2.7. (Mach number = v/vsound. Assume the speed of sound is 343 m/s.)
(d) The space shuttle, traveling 27,000 km/h.
(e) An electron traveling 30 cm in 2 ns.
(f) A proton traveling across a nucleus (10-14 m) in 0.38 ✕ 10-22 s.

Answer 1

a)
`\beta = 1.111* 10^(-7)`, b)
`\beta = 9* 10^(-7)`, c)
`\beta = 3.087* 10^(-6)`, d)
`\beta = 2.5* 10^(-5)`, e)
`\beta = 0.5`, f)
`\beta = 0.877`

Step-by-step explanation:

From relativist physics we know that
`c` is the symbol for the speed of light, which equal to approximately 300000 kilometers per second. (300000000 meters per second).

a) A car traveling 120 kilometers per hour:

At first we convert the car speed into meters per second:

`v = \left(120\,(km)/(h) \right)* \left(1000\,(m)/(km) \right)* \left((1)/(3600)\,(h)/(s) \right)`

`v = 33.333\,(m)/(s)`

The ratio
`\beta` is now calculated: (
`v = 33.333\,(m)/(s)`,
`c = 3* 10^(8)\,(m)/(s)`)

`\beta = (33.333\,(m)/(s) )/(3* 10^(8)\,(m)/(s) )`

`\beta = 1.111* 10^(-7)`

b) A commercial jet airliner traveling 270 meters per second:

The ratio
`\beta` is now calculated: (
`v = 270\,(m)/(s)`,
`c = 3* 10^(8)\,(m)/(s)`)

`\beta = (270\,(m)/(s) )/(3* 10^(8)\,(m)/(s) )`

`\beta = 9* 10^(-7)`

c) A supersonic airplane traveling Mach 2.7:

At first we get the speed of the supersonic airplane from Mach's formula:

`v = Ma\cdot v_(s)`

Where:

`Ma` - Mach number, dimensionless.

`v_(s)` - Speed of sound in air, measured in meters per second.

If we know that
`Ma = 2.7` and
`v_(s) = 343\,(m)/(s)`, then the speed of the supersonic airplane is:

`v = 2.7\cdot \left(343\,(m)/(s) \right)`

`v = 926.1\,(m)/(s)`

The ratio
`\beta` is now calculated: (
`v = 926.1\,(m)/(s)`,
`c = 3* 10^(8)\,(m)/(s)`)

`\beta = (926.1\,(m)/(s) )/(3* 10^(8)\,(m)/(s) )`

`\beta = 3.087* 10^(-6)`

d) The space shuttle, travelling 27000 kilometers per hour:

At first we convert the space shuttle speed into meters per second:

`v = \left(27000\,(km)/(h) \right)* \left(1000\,(m)/(km) \right)* \left((1)/(3600)\,(h)/(s) \right)`

`v = 7500\,(m)/(s)`

The ratio
`\beta` is now calculated: (
`v = 7500\,(m)/(s)`,
`c = 3* 10^(8)\,(m)/(s)`)

`\beta = (7500\,(m)/(s) )/(3* 10^(8)\,(m)/(s) )`

`\beta = 2.5* 10^(-5)`

e) An electron traveling 30 centimeters in 2 nanoseconds:

If we assume that electron travels at constant velocity, then speed is obtained as follows:

`v = (d)/(t)`

Where:

`v` - Speed, measured in meters per second.

`d` - Travelled distance, measured in meters.

`t` - Time, measured in seconds.

If we know that
`d = 0.3\,m` and
`t = 2* 10^(-9)\,s`, then speed of the electron is:

`v = (0.3\,m)/(2* 10^(-9)\,s)`

`v = 1.50* 10^(8)\,(m)/(s)`

The ratio
`\beta` is now calculated: (
`v = 1.5* 10^(8)\,(m)/(s)`,
`c = 3* 10^(8)\,(m)/(s)`)

`\beta = (1.5* 10^(8)\,(m)/(s) )/(3* 10^(8)\,(m)/(s) )`

`\beta = 0.5`

f) A proton traveling across a nucleus (10⁻¹⁴ meters) in 0.38 × 10⁻²² seconds:

If we assume that proton travels at constant velocity, then speed is obtained as follows:

`v = (d)/(t)`

Where:

`v` - Speed, measured in meters per second.

`d` - Travelled distance, measured in meters.

`t` - Time, measured in seconds.

If we know that
`d = 10^(-14)\,m` and
`t = 0.38* 10^(-22)\,s`, then speed of the electron is:

`v = (10^(-14)\,m)/(0.38* 10^(-22)\,s)`

`v = 2.632* 10^(8)\,(m)/(s)`

The ratio
`\beta` is now calculated: (
`v = 2.632* 10^(8)\,(m)/(s)`,
`c = 3* 10^(8)\,(m)/(s)`)

`\beta = (2.632* 10^(8)\,(m)/(s) )/(3* 10^(8)\,(m)/(s) )`

`\beta = 0.877`