A particle moves along the x-axis with the velocity given by v(t)=3t/(1+t^2) for t >or equal to 0. When t=0, the particle is at the point (4,0). 1. Determine the maximum velocity for the particle. Justify your answer. 2. Determine the position of the particle at any time t. 3. Find the limit of the velocity as t->infinity. 4. Find the limit of the position as t->infinity.
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dx = vdt use this..this little thing is supposed to do wonders for you ! :D
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Just started learning it. It's not my best subject. xD
the definition of velocity is v=dx/dt
1. a(t) = 3(1-t^2)/(1+t^2)^2 max v at a=0, or t=1 v(1) = 3/2 2. x(t) = 3/2 log(1+t^2)+C at t=0, x=4, so 4 = 3/2 log(1+0) + C C = 4 x(t) = 3/2 log(1+t^2) + 4 3. v(t) → 0 as t→∞ 4. x(t) → ∞ that's what I've got so far. Does that make sense?
you sure differentiated right in the 1st part.. rest all seem perfect! :)
At least someone agrees with it so far.
i missed a question mark there :P i meant you sure differentiated right in the 1st part ???? :D cause i see that wrong..please confirm..
Could somebody please check this to make sure it makes sense? 1. dv/dt = [(1+t^2)(3) - 3t(2t)] / (t^2+1)^2 =[3 t^2 + 3 - 6 t^2 ] /(t^2+1)^2 zero when t= 1 (since t always >/=0) so the max v is at t = 1 then at t = 1 v(1) = 3/2 2. x(t) = 3/2 log(1+t^2)+C at t=0, x=4, so 4 = 3/2 log(1+0) + C C = 4 x(t) = 3/2 log(1+t^2) + 4 3. as t --> oo, v--> 3t/t^2 = 3/t = 0 4. as t -->oo, x --> (3/2)t^2 = oo