Between t = 0 and t = t0, a rocket moves straight upward with an acceleration given by a(t) = a − bt1 /2 , where a and b are constants. (a) if x is in meters and t is in seconds, what are the units of a and b? (b) if the rocket starts from rest, how does the velocity vary between t = 0 and t = t0? (c) if its initial position is zero, what is the rocket’s position as a function of time during this same time interval?

a) a= [m/s²]     B = [m / ]     b) v = A t - 2/3 B √t³  

Explanation:

The equation gives us for acceleration is a (t) = A - B √t  

a) The unit of constants

Since the function is the acceleration and you have units of [m / s²] the quantities the right must have the same units, therefore, the counter A is in [m / s²] and the other term is B for the time

   B t ½ = [m / s²]

   B [s ½] = [m / s²]

   B = [m / s²] ] = [m] / [  s²] = [m] / [  ]

   B = [m / ]

b) To calculate the rocket speed we must use the definition of acceleration and integrate

    a = dv / dt

    dv = a dt

    ∫ dv = ∫ (A - B √t) dt

    v = A (t) - B  √t³   / 3/2

    v = At - 2/3 B  √t³

We evaluated entered the initial point where the zero velocity equals a zero time and the final point with velocity v and time t

     v = A t - 2/3 B √t³  

c) To find the position we repeat the same proceeding for speed

     v = dx / dt

     dx = v dt

     dx = / v dt

     ∫ dx = ∫ (A t - 2/3 B √t³ ) dt

     x =  At² / 2 - 2/3 B √t⁵  / 5/2

     x = ½ A t² - 2/3 2/5 B √t⁵

Evaluating from the start point with zero position for a time of zero or to the end point with position x at time t

      x = ½ A t² - 4/15 B √t⁵

the order for strong to weak forces such as

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force

the fundamental forces are those forces that act between to objects or element to make interactions. there are four types of fundamental forces such as strong and weak nuclear forces (due to protons and electrons), electromagnetic force (due to the motion of electrons) and gravitational force (due to orbital movement of the planet). if these forces are arranged on the basis of their maximum strength then the order is

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force