A 2090-kg test rocket is launched vertically from the launch pad. Its fuel (of negligible mass) provides a thrust force so that its vertical velocity as a function of time is given by v(t) =At+Bt^2 , where A and B are constants and time is measured from the instant the fuel is ignited. The rocket has an upward acceleration of 1.50m/s 2 at the instant of ignition and, 1.00 s later, an upward velocity of 2.00 m/s. (a) Determine A and B , including their SI units. (b) At 4.00 s after fuel ignition, what is the acceleration of the rocket, and (c) what thrust force does the burning fuel exert on it, assuming no air resistance? Express the thrust in newtons and as a multiple of the rocket’s weight. (d) What was the initial thrust due to the fuel?

a) A = 1.50 m / s²,  B = 1.33 m/s³,  b) a = 12.1667 m / s²,

c)  I = M (1.5 t + 1.333 t²) ,  d)  ΔI = M 2.833   N

Explanation:

In this exercise give the expression for the speed of the rocket

         v (t) = A t + B t²

and the initial conditions

         a = 1.50 m / s² for t = 0 s

         v = 2.00 m / s for t = 1.00 s

a) it is asked to determine the constants.

Let's look for acceleration with its definition

         a =

         a = A + 2B t

we apply the first condition t = 0 s

         a = A

         A = 1.50 m / s²

we apply the second condition t = 1.00 s

          v = 1.5 1 + B 1²

          2 = 1.5 + B

          B = 2 / 1.5

          B = 1.33 m/s³

the equation remains

           v = 1.50 t + 1.333 t²

b) the acceleration for t = 4.00 s

           a = 1.50 + 1.333 2t

           a = 1.50 + 2.666 4

           a = 12.1667 m / s²

c) The thrust

           I = ∫ F dt = p_f - p₀

Newton's second law

          F = M a

          F = M (1.5 + 2 1.333 t) dt

we replace and integrate

         I = M ∫ (1.5 + 2.666 t) dt

         I = 1.5 t + 2.666 t²/2

         I = M (1.5 t + 1.333 t²) + cte

in general the initial rockets with velocity v = 0 for t = 0, where we can calculate the constant

         cte = 0

         I = M (1.5 t + 1.333 t²)

d) the initial push

For this we must assume some small time interval, for example between

t = 0 s and t = 1 s

        ΔI = I_f - I₀

        ΔI = M (1.5 1 + 1.333 1²)

        ΔI = M 2.833   N