Answer:
the average speed of the particles in terms of the speed of light is 0.81c
Explanation:
Given the data in the question;
= 256.2 ps = 256.2 × 10⁻¹² s
speed of light c = 2.99 × 10⁸ m/s
d = 10.7 cm = 0.107 m
we know that; Average speed v = d/t ------- let this be equation 1
Also, given that 256.2 ps is the lifetime of particle X frame, proper time will be;
t = Y = / √( 1 - ) --------- let this be equation 2
Next, we input equation 2 into equation 1'
v = d / [ / √( 1 - ) ]
v = d/[ √( 1 - ) ]
v/d = √( 1 - )
we square both sides
( v/d )² = (√( 1 - ) )²
v²²/d² = 1 -
v²²/d² + = 1
v²( ²/d² + ) = 1
v²( (²c² + d²)/d²c² ) = 1
∴
v² = (d²c²) / (²c² + d²)
v = √[ (d²c²) / (²c² + d²) ]
v = (dc) / √(²c² + d²)
so we substitute
v = (0.107 m × c) / √( (256.2 × 10⁻¹² s)²(2.99 × 10⁸ m/s)² + (0.107 m )²)
v = 0.107c / √( 0.00586814 + 0.011449 )
v = 0.107c / √( 0.01731714 )
v = 0.107c / 0.1315946
v = 0.81c
Therefore, the average speed of the particles in terms of the speed of light is 0.81c