4.15 A small 7.40- k g rocket burns fuel that exerts a time-varying upward force

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?

Explanation / Answer

F = Bt^2 + A at t=0, f=101N
at t=1.6, f=200N

{rearrange for A}

A = F - Bt^2 so at t=0 F = 101

{sub 0 for t and 101 for F to find A}

therefore: A = 101 - B(0)^2 = 101

now

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

B = (F - 101) / (t^2) = (200-101) / (1.6)^2

Therefore B = 38.67 roughly

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

F = Bt^2 + A

F should = 101, when t = 0

38.67(0)^2 + (101) = 101 that works.

F should = 200, when t=1.6

38.67(1.6)^2 + 101 = 199.9952 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 5.7kg, therefore the force due to gravity at sea level is 9.8ms/s*5.7kg = 55.86N and the upwards force on the rocket at t=0, we already know is 101N
so net force = 101N - 55.86N = 45.14N

(d) acceleration = net force / mass
therefore a = 45.14N / 5.7kg = 7.92m/s^-2

{use original formula}
(e) F = 38.67(5)^2 + 101 = 1067.75N
so net = (1067.75-55.86)N = 1011.89N

(f) again, a = f/m therefore a = 1011.89N / 5.7kg
therefore a = 177.52ms^-2

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

F = 1067.75N , do a = (1067.75N) / (5.7kg) = 187.32m/s^-2

Please dont rely 100% on my answers, as doing is everything when learning physics, also I can't guarantee I haven't made some mistakes, since I didnt have much time to do this.

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