Question

4.15

A small 7.40-kg rocket burns fuel that exerts a time-varying upward force on the rocket. This force obeys the equation F=A+Bt2. Measurements show that at t=0, the force is 104.0N , and at the end of the first 1.50s , it is 193.0N .

Find the constant A

Find the constants B

Find the net force on this rocket the instant after the fuel ignites.

Find the acceleration of this rocket the instant after the fuel ignites.

Find the net force on this rocket 4.90s after fuel ignition.

Find the acceleration of this rocket 4.90s after fuel ignition.

Suppose you were using this rocket in outer space, far from all gravity. What would its acceleration be 4.90s after fuel ignition?

Answer 1

F = Bt^2 + A at t=0, f=104N
at t=1.5, f=193 N

{rearrange for A}

A = F - Bt^2 so at t=0 F = 104

{sub 0 for t and 104 for F to find A}

therefore: A = 104 - B(0)^2 = 104

now

{sub 200 for F)
F = Bt^2 + 101 therefore 193 = Bt^2 + 104
{ finally rearrange to solve for B}

B = (F - 104) / (t^2) = (193-104 ) / (1.5)^2

Therefore B = 39.55 roughly

We can check its right by subbing values to the original equation:

F = Bt^2 + A

F should = 104, when t = 0

39.55(0)^2 + (104) = 104 that works.

F should = 193, when t=1.5

39.55(1.5)^2 + 104 = 193 on calculator, which sounds spot on to me, as my answer for Was rounded anyway.

Hope this helps. You just need to get used to handling formulas and rearranging them.


now for (c)

the net force the moment the fuel ignites?

the mass of the rocket is 7.4 kg, therefore the force due to gravity at sea level is 9.8ms/s*7.4kg = 72.52 N and the upwards force on the rocket at t=0, we already know is 104N
so net force = 104N - 72.52N =31.48N

(d) acceleration = net force / mass
therefore a = 31.48N / 7.4kg =4.25m/s^-2

{use original formula}
(e) F = 39.55(4.9)^2 + 104 = 1053.59N
so net = (1067.75-72.52)N = 981.07 N

(f) again, a = f/m therefore a = 981.07N /7.4kg
therefore a = 133.577ms^-2

(g) in space we can neglect gravitational force, and we have:

F = 1053.59N , do a = (1053.59N) / (7.4kg) = 142.37m/s^-2