Expert: Paul Klarreich - 4/6/2008

Question
I am having some trouble here with this problem. I am not sure how to start or what to do at all.

A particle moves along the y-axis with velocity given by v(t)=tsin(t^2) for t>=0.

a)which direction(up or down) is the particle moving at time t=1.5?
b)Find the acceleration of the particle at time t=1.5. Is the velocity of the particle increasing at t=1.5?
c)Given that y(t) is the position of the particle at time t and that y(0)=3, find y(2).
d)Find the total distance traveled by the particle from t=0 to t=2.

Answer
Questioner:   Shanye
Category:  Calculus
Private:  No
 
Subject:  Motion
Question:  I am having some trouble here with this problem. I am not sure how to start or what to do at all.

A particle moves along the y-axis with velocity given by v(t)=tsin(t^2) for t>=0.

a)which direction(up or down) is the particle moving at time t=1.5?
b)Find the acceleration of the particle at time t=1.5. Is the velocity of the particle increasing at t=1.5?
c)Given that y(t) is the position of the particle at time t and that y(0)=3, find y(2).
d)Find the total distance traveled by the particle from t=0 to t=2.
........................................
Hi, Shanye,

If v(t) is given, then:

v(t) > 0 means 'going up'.
v(t) < 0 means 'going down'.
The acceleration, a(t) is v'(t)
if a(t) > 0, velocity is increasing.
The position, y(t) is the integral of v(t).

With that stuff in mind,

a)which direction(up or down) is the particle moving at time t=1.5?

v(1.5) = 1.5 sin (2.25)

Put your calc into radian mode and compute that. If >0, .... well, you'll know what to do.
.......................
b)Find the acceleration of the particle at time t=1.5. Is the velocity of the particle increasing at t=1.5?

a(t) = v'(t) = sin(t^2) + 2t^2 cos(t^2)  << Product rule.

a(1.5) = sin(2.25) + 2(2.25) cos(2.25)

Same drill.
....................
c)Given that y(t) is the position of the particle at time t and that y(0)=3, find y(2).

Integrate:

{
| t sin(t^2) dt,  using  u = t^2,  du = 2t dt,  t dt = du/2
}

{
| sin u du/2 = -(1/2) cos u + C = -(1/2) cos (t^2) + C.
}

Now if  y(0) = 3, we can find C.

y(0) = -1/2 cos(0) + C = 3
-1/2 + C = 3
C = 7/2

y(t) = -(1/2) cos (t^2) + 7/2.

y(2) = - 1/2(4) + 7/2 = -2 + 7/2 = 3/2


d)Find the total distance traveled by the particle from t=0 to t=2.

That's just  y(2) - y(0)