Question

A diwali rocket is ejecting 50 g of gases at velocity of 400m/s .the accelerating force on the rocket will be...???

Answer 1

`"20 N"`

Explanation:

I think that you mistyped the question.

More specifically, I think that the rocket actually ejects `"50 g"` of gases per second at a velocity of `"400 m/s"`.

That would make more sense, since `"50 g"` of gases in total would not make for a very impressive rocket.

So, the idea here is that the constant rate at which the gases are being ejected corresponds to the rocket's movement at constant speed.

If an object moves at constant speed, its acceleration, which is the derivative of the speed with respect to time, is equal to zero.

`d/dt(v) = a = 0 <=> v = "const"`

The speed of the rocket is constant, but its mass is not. This implies that its momentum will not be constant.

Assuming that the rocket flies straight upward, momentum is defined as

`P = m * v`

The momentum of the rocket changes with respect to time because the mass of the rocket changes with respect to time.

According to Newton's Second Law, the rate of change of the momentum of an object is proportional to the force that's acting on it.

In your case, this force will be the ccelerating force, `F`

`d/dt(P) = d/dt(m * v)`

`F = d/dt(m * v)`

Use the product rule to differentiate this function

`F = d/dt(m) * v + m * underbrace(d/dt(v))_(color(blue)(=0))`

This means that

`F = v * underbrace(d/dt(m))_(color(blue)(="50 g/s"))`

To get the result in Newtons, convert the rate of change of the mass of the gases to `"kg/s"`

`50color(red)(cancel(color(black)("g")))/"s" * "1 kg"/(1000color(red)(cancel(color(black)("g")))) = "0.050 kg/s"`

The accelerating force will thus be

`F = 400"m"/"s" * 0.050"kg"/"s" = 20 "kg m"/"s"^2 = color(green)("20 N")`