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Help making an equation for motion?
I have trouble understanding the answer of the following question:
A toy rocket of mass 0.2kg is projected vertically upwards from rest by means of a force which decreases uniformly in 2 seconds from 2kg wt to zero and thereafter ceases.
Write equations of motion for the rocket for the case t < 2.
dV/dt = (9 - 5t)g.
Why is that so? Thanks!
F = ma = m dv/dt
You are told the propelling force decreases linearly in time, so it is of the form F = a + bt, and b is negative because it is decreasing.
At t = 0, the force is "2 kg wt". I assume that means 2 kg * g. Plug in t = 0 and set it equal to 2g:
F(0) = a + b*0 = a = 2g. So now you know the constant a.
You are told that the force is 0 at t = 2.
F(2) = a + 2b = 0. Solving, b = -a/2 = -2g/2 = -g
Now you know b, so F = a + bt = 2g - gt
One more piece: Don't forget gravity, which acts downward and has magnitude mg. So the total force on the rocket is 2g - gt - mg. This equal to mass times acceleration and that gives you your equation of motion:
2g - gt - mg = m dv/dt
Since you know m = 0.2, you can substitute that to get
2g - gt - 0.2 g = 0.2 dv/dt
1.8g - gt = 0.2 dv/dt
And that is your equation of motion. If you multiply both sides by 5 to just get dv/dt on the right, you'll have the equation you were told is the answer.
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