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 total distance that martina travels that day is

(distance from home to mom's office) plus (distance from mom's office to school)

plus (distance from school to library) plus (distance from library to friend's house)

plus (distance from friend's house to martina's house).

(we don't know what any of these distance actually is.)

d. platinum would have the greatest temperature increase because it has the lowest specific heat.