The Proper life time of a particle is given as t=1ns, and we need to determine the distance traveled by the particle before it decays. The particle is traveling at a speed of v=3c/5, where c is the speed of light.

To find the distance traveled by the particle, we can use the formula for distance, which is given by the product of speed and time.

Formula:
Distance = Speed × Time

Given:
Proper life time (t) = 1 ns
Speed (v) = 3c/5

Solution:

Step 1: Convert the time into seconds
Given time is 1 ns.
1 nanosecond (ns) = 1 × 10^(-9) seconds (s)

Step 2: Substitute the values into the formula
Distance = Speed × Time
Distance = (3c/5) × (1 × 10^(-9) s)

Step 3: Simplify the expression
To simplify the expression, we need to substitute the value of the speed of light.
The speed of light (c) is approximately 3 × 10^8 m/s.

Distance = (3 × 3 × 10^8 m/s / 5) × (1 × 10^(-9) s)

Simplifying further,
Distance = (9 × 10^8 m/s / 5) × (1 × 10^(-9) s)
Distance = (1.8 × 10^8 m/s) × (1 × 10^(-9) s)

Step 4: Multiply the values
To multiply the values, we can multiply the coefficients and add the exponents of 10.

Distance = 1.8 × 10^(8+(-9)) m
Distance = 1.8 × 10^(-1) m

Step 5: Rewrite the distance in a proper format
The distance is 1.8 × 10^(-1) m, which can be written as 0.18 m in decimal notation.

Therefore, the distance traveled by the particle before it decays is 0.18 m.