Respuesta :
Answer:
a) [tex]\beta = 1.111\times 10^{-7}[/tex], b) [tex]\beta = 9\times 10^{-7}[/tex], c) [tex]\beta = 3.087\times 10^{-6}[/tex], d) [tex]\beta = 2.5\times 10^{-5}[/tex], e) [tex]\beta = 0.5[/tex], f) [tex]\beta = 0.877[/tex]
Explanation:
From relativist physics we know that [tex]c[/tex] 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:
[tex]v = \left(120\,\frac{km}{h} \right)\times \left(1000\,\frac{m}{km} \right)\times \left(\frac{1}{3600}\,\frac{h}{s} \right)[/tex]
[tex]v = 33.333\,\frac{m}{s}[/tex]
The ratio [tex]\beta[/tex] is now calculated: ([tex]v = 33.333\,\frac{m}{s}[/tex], [tex]c = 3\times 10^{8}\,\frac{m}{s}[/tex])
[tex]\beta = \frac{33.333\,\frac{m}{s} }{3\times 10^{8}\,\frac{m}{s} }[/tex]
[tex]\beta = 1.111\times 10^{-7}[/tex]
b) A commercial jet airliner traveling 270 meters per second:
The ratio [tex]\beta[/tex] is now calculated: ([tex]v = 270\,\frac{m}{s}[/tex], [tex]c = 3\times 10^{8}\,\frac{m}{s}[/tex])
[tex]\beta = \frac{270\,\frac{m}{s} }{3\times 10^{8}\,\frac{m}{s} }[/tex]
[tex]\beta = 9\times 10^{-7}[/tex]
c) A supersonic airplane traveling Mach 2.7:
At first we get the speed of the supersonic airplane from Mach's formula:
[tex]v = Ma\cdot v_{s}[/tex]
Where:
[tex]Ma[/tex] - Mach number, dimensionless.
[tex]v_{s}[/tex] - Speed of sound in air, measured in meters per second.
If we know that [tex]Ma = 2.7[/tex] and [tex]v_{s} = 343\,\frac{m}{s}[/tex], then the speed of the supersonic airplane is:
[tex]v = 2.7\cdot \left(343\,\frac{m}{s} \right)[/tex]
[tex]v = 926.1\,\frac{m}{s}[/tex]
The ratio [tex]\beta[/tex] is now calculated: ([tex]v = 926.1\,\frac{m}{s}[/tex], [tex]c = 3\times 10^{8}\,\frac{m}{s}[/tex])
[tex]\beta = \frac{926.1\,\frac{m}{s} }{3\times 10^{8}\,\frac{m}{s} }[/tex]
[tex]\beta = 3.087\times 10^{-6}[/tex]
d) The space shuttle, travelling 27000 kilometers per hour:
At first we convert the space shuttle speed into meters per second:
[tex]v = \left(27000\,\frac{km}{h} \right)\times \left(1000\,\frac{m}{km} \right)\times \left(\frac{1}{3600}\,\frac{h}{s} \right)[/tex]
[tex]v = 7500\,\frac{m}{s}[/tex]
The ratio [tex]\beta[/tex] is now calculated: ([tex]v = 7500\,\frac{m}{s}[/tex], [tex]c = 3\times 10^{8}\,\frac{m}{s}[/tex])
[tex]\beta = \frac{7500\,\frac{m}{s} }{3\times 10^{8}\,\frac{m}{s} }[/tex]
[tex]\beta = 2.5\times 10^{-5}[/tex]
e) An electron traveling 30 centimeters in 2 nanoseconds:
If we assume that electron travels at constant velocity, then speed is obtained as follows:
[tex]v = \frac{d}{t}[/tex]
Where:
[tex]v[/tex] - Speed, measured in meters per second.
[tex]d[/tex] - Travelled distance, measured in meters.
[tex]t[/tex] - Time, measured in seconds.
If we know that [tex]d = 0.3\,m[/tex] and [tex]t = 2\times 10^{-9}\,s[/tex], then speed of the electron is:
[tex]v = \frac{0.3\,m}{2\times 10^{-9}\,s}[/tex]
[tex]v = 1.50\times 10^{8}\,\frac{m}{s}[/tex]
The ratio [tex]\beta[/tex] is now calculated: ([tex]v = 1.5\times 10^{8}\,\frac{m}{s}[/tex], [tex]c = 3\times 10^{8}\,\frac{m}{s}[/tex])
[tex]\beta = \frac{1.5\times 10^{8}\,\frac{m}{s} }{3\times 10^{8}\,\frac{m}{s} }[/tex]
[tex]\beta = 0.5[/tex]
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:
[tex]v = \frac{d}{t}[/tex]
Where:
[tex]v[/tex] - Speed, measured in meters per second.
[tex]d[/tex] - Travelled distance, measured in meters.
[tex]t[/tex] - Time, measured in seconds.
If we know that [tex]d = 10^{-14}\,m[/tex] and [tex]t = 0.38\times 10^{-22}\,s[/tex], then speed of the electron is:
[tex]v = \frac{10^{-14}\,m}{0.38\times 10^{-22}\,s}[/tex]
[tex]v = 2.632\times 10^{8}\,\frac{m}{s}[/tex]
The ratio [tex]\beta[/tex] is now calculated: ([tex]v = 2.632\times 10^{8}\,\frac{m}{s}[/tex], [tex]c = 3\times 10^{8}\,\frac{m}{s}[/tex])
[tex]\beta = \frac{2.632\times 10^{8}\,\frac{m}{s} }{3\times 10^{8}\,\frac{m}{s} }[/tex]
[tex]\beta = 0.877[/tex]
