calculus
posted by Tiffany
4. A particle starts at the point (5, 0) at t = 0 and moves along the xaxis in such a way that at time t > 0 the velocity is given by v(t)=t/(1+t^2)
a. Determine the maximum velocity attained by the particle. Justify your answer.
b. Determine the position of the particle at t = 6.
c. Find the limiting value of the velocity as t increases without bound.
d. Does the particle ever pass the point (500,0)? Explain.

Damon
dv/dt = [ (1+t^2) t(2t) ] / (1+t^2)^2
= (1t^2) /(1+t^2)^2
zero when
t = + or  1
when t = +1
v = 1/2
when t = 1 that was before we started
so v max is 1/2 at t = 1
integrate dx = v dt to get x(6), remember to add 5
(1/2)ln(t^2+1) from t = 0 to 6
it is 0 when t = 0
so
x = 5 + .5 ln 37 when x = 6
looks like 1/t for large t, so approaches 0
sure it reaches all +x
because as t > oo
x = 5 + ln (oo)
Respond to this Question
Similar Questions

Calculus
a particle starts at time t = 0 and moves along the x axis so that its position at any time t>= 0 is given by x(t) = ((t1)^3)(2t3) a.find the velocity of the particle at any time t>= 0 b. for what values of t is the velocity … 
Calculus
A particle moves along a line so that at any time t its position is given by x(t)=2(pi)t+cos(2(pi)t). Determine the particle's maximum velocity. 
calculus
a particle moves along the xaxis in such a way that its acceleration at any time t is given by a(t)= 6t  18. At time t = 0, the velocity v(t) = 24. and at time t = 1, the position x(t) = 20 i have to find the expression for the velocity … 
Calculus
A particle moves along the xaxis with velocity v(t) = (t3)² + 5 for [0,6]. a) Find the average velocity of this particle during the interval [0,6]. b) Find a time t* ∈ [0,6] such that the velocity at time t* is equal to the … 
calculus
5. A particle moves along the xaxis in such a way that its position at time t is given by x=3t^416t^3+24t^2 for 5 ≤ t ≤ 5. a. Determine the velocity and acceleration of the particle at time t. b. At what values of t … 
calculus
5. A particle moves along the y – axis with velocity given by v(t)=tsine(t^2) for t>=0 . a. In which direction (up or down) is the particle moving at time t = 1.5? 
Calculus
A particle moves along the xaxis 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 … 
ap calculus
a particle moves along the x axis in such a way that its acceleration at time t, t>0 , is given by x(t)= (ln x)^2. at what value of t does the velocity of the particle attain its maximum 
Calculus
A particle starts at the point (5,0) at t=0 and moves along the xaxis in such a way that at time t>0 its velocity v(t) is given by v(t)= t/(1+t^2). a). Determine the maximum velocity of the particle. Justify your answer. b). Determine … 
Calculus
The Question: A particle moves along the Xaxis so that at time t > or equal to 0 its position is given by x(t) = cos(√t). What is the velocity of the particle at the first instance the particle is at the origin?