Expert: Paul Klarreich - 10/6/2008

Question
A particle moves along a line so that at any time t its position is given by x(t)=2(pi)t+cos2(pi)t.
(a) Find the velocity at time t.
(b) Find the acceleration at time t.
(c) What are all values of t, 0<(or equal to)t<(or equal to)3, for which the particle is at rest?
(d) What is the maximum velocity?

I know that I need to find the derivative for the velocity at time t, but I am unsure of how to find acceleration.  Also, how do I get to find the maximum velocity?  I am in AP Calc now, and we are studying derivatives and relative extrema.

Answer
Questioner:   Gina
Category:  Calculus
Private:  No
 
Subject:  Derivatives
Question:  A particle moves along a line so that at any time t its position is given by x(t)=2(pi)t+cos2(pi)t.
(a) Find the velocity at time t.
(b) Find the acceleration at time t.
(c) What are all values of t, in  0 <=  t <= 3, for which the particle is at rest?
(d) What is the maximum velocity?

I know that I need to find the derivative for the velocity at time t, but I am unsure of how to find acceleration.  Also, how do I get to find the maximum velocity?  I am in AP Calc now, and we are studying derivatives and relative extrema.
......................................
Hi, Gina,

Thank you for reading my instructions; you wouldn't believe how many people pay no attention.

If x(t) = 2 pi t+  cos(2 pi t),
then v(t) is the derivative of x(t) and  a(t) is the derivative of v(t) [the second derivative.]

So  v(t) = x'(t) =  2 pi - 2pi sin(2 pi t)

a(t) = v'(t) = x''(t) =  - 4 pi^2 cos (2 pi t)

Now we can get to work.
......................
(c) What are all values of t, in  0 <=  t <= 3, for which the particle is at rest?

AT REST means  v = 0.

Set 2 pi - 2pi sin(2 pi t) = 0 and solve.

2 pi = 2pi sin(2 pi t)

1 = sin(2 pi t)

Now  sin theta = 1 if  theta =  pi/2, 5pi/2, 9pi/2,...

Set 2 pi t = pi/2, 5pi/2, 9pi/2, ...
Cancel the pi,
2t = 1/2, 5/2, 9/2, ...

AND  t = 1/4, 5/4, 9/4, and some larger numbers.

...................

(d) What is the maximum velocity?

You achieve max velocity when  a(t) = 0.

Set - 4 pi^2 cos (2 pi t) = 0

cos (2 pi t) = 0

Now cos theta = 0 if  theta = pi/2, 3pi/2, and more values that we don't need right now.

Set  2 pi t = pi/2,  3pi/2

Then  t = 1/4, 3/4.

That's WHEN you get your max velocity.  What is it?  Plug them in:

v(t) =  2 pi - 2pi sin(2 pi t)

v(1/4) =  2 pi - 2pi sin(2 pi (1/4))
v(1/4) =  2 pi - 2pi sin(pi/2)
v(1/4) =  2 pi - 2pi (1) = 0  << That is NOT your max velocity.

v(3/4) =  2 pi - 2pi sin(2 pi (3/4))
v(3/4) =  2 pi - 2pi sin(3pi/4))
v(3/4) =  2 pi - 2pi(-1) = 4 pi  << That should be your max.

BUT be a good girl and check out the endpoints:

t = 0:  v(0) = 2 pi(0) - 2pi sin(0) = 0.

t = 3:  v(0) = 2 pi(3) - 2pi sin(6pi) = 2pi( 3 - 1) = 4 pi, also a max.