A sample of polonium-210 was left for 414 days. BA
After this time it had a mass of 1.45 x 10^-4 g
The half-life of polonium-210 is 138 days.
Calculate the initial mass of the sample

PO(1) = PO(0) / 2       1 refers to 1 half life of PO(0)

PO(2) = PO(1) / 2 = P(0) / 4        amount of PO left after 2 half-lives

PO(3) = PO(2) / 2 = PO(0) / 8     amount of PO left after 3 half-lives

414 da / 138 da = 3       3 half-lives pass in 414 da

PO(0) = 8 PO(3) = 8 * 1.45E-4 g = 1.16E-3 g = .00116 g  after 414 days        

145.8 m

Explanation:

M = mass of the runaway train car = 15,000 kg

V₀ = initial speed of the train = 5.4 m/s

V = final speed of the train = 0 m/s

F = force applied to bring the car to rest = 1500 N

d = distance traveled before stopping

Using work-change in kinetic energy theorem

Work done by applied force = Change in kinetic energy

- F d = (0.5) M (V² - V₀²)

- (1500) d = (0.5) (15000) (0² - 5.4²)

d = 145.8 m