People hoping to travel to other worlds are faced with huge challenges. one of the biggest is the time required for a journey. the nearest star is 4.1×1016m away. suppose you had a spacecraft that could accelerate at 1.3 g for half a year, then continue at a constant speed. (this is far beyond what can be achieved with any known technology.) part a how long would it take you to reach the nearest star to earth?

The travel would take 6.7 years.

Explanation:

The equation for an object moving in a straight line with acceleration is:

x = x0 + v0 t + 1/2a*t²

where:

x = position at time t

x0 = initial position

v0 = initial velocity

a = acceleration

t = time

In a movement with constant speed, a = 0 and the equation for the position will be:

x = x0 + v t

where v = velocity

Let´s calculate the position from the Earth after half a year moving with an acceleration of 1.3 g = 1.3 * 9.8 m/s² = 12.74  m/s²:

Seconds in half a year:

1/2 year = 1.58 x 10⁷ s

x = 0 m + 0 m/s + 1/2 * 12.74 m/s² * (1.58 x 10⁷ s)²  = 1.59 x 10¹⁵ m

Now let´s see how much time it takes the travel to the nearest star after this half year.

The velocity will be the final velocity achived after the half-year travel with an acceleration of 12.74 m/s²

v = v0 + a t

Since the spacecraft starts from rest, v0 = 0

v = 12.74 m/s² * 1.58 x 10⁷ s = 2.01 x 10 ⁸ m/s

Using the equation for position:

x = x0 + v t

4.1 x 10¹⁶ m = 1.59 x 10¹⁵ m + 2.01 x 10 ⁸ m/s * t

(4.1 x 10¹⁶ m - 1.59 x 10¹⁵ m) / 2.01 x 10 ⁸ m/s = t

t = 2.0 x 10⁸ s * 1 year / 3.2 x 10 ⁷ s = 6.2 years.

The travel to the nearest star would take 6.2 years + 0.5 years = 6.7 years.

the body's situation won't change the forces are equal opposit and on the same body

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.