A uranium nucleus 238U may stay in one piece for billions of years, but sooner or later it decays into an α particle of mass 6.64 x 10^−27 kg and 234Th nucleus of mass 3.88 x 10^−25 kg, and the decay process itself is extremely fast (it takes about 10−20 s). Suppose the uranium nucleus was at rest just before the decay.
a) If the α particle is emitted at a speed of 5.24 x 10^6 m/s, what would be the recoil speed of the thorium nucleus?
Answer in units of m/s

The recoil speed is - 8.9604. m/s.

Explanation: According to the Third Law of Newton, every action has an oppsite and equal reaction, and the Second Law of Newton, Force=mass·acceleration. Acceleration is a variation in velocity by any given time, so Force = mass·velocity·time.

Combining the two laws, there is : m1·v1 = - m2·v2. This is the Law of Conservation of Momentum.

Substituting and calculating:

v2 = - () · v1

v2 = - · 5.24.

v2 = - 8.9604.

The recoil speed of the thorium nucleus is - 8.9604.m/s.

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.