A particle P is moving along the x-axis. Its position at time t seconds is given by: (1 Viewer)

[come in]

A particle P is moving along the x-axis. Its position at time t seconds is given by:

x = 2sint - t , where t >= 0

[This is the order of the questions. I did 1-5 already, I just need 6]

1) Find an expression for the velocity of the particle.
ANS: V = 2cost - 1

2) In what direction is the particle moving at t = 0
ANS: right

3) Determine when the particle first comes to rest.
ANS: t = pi/3 seconds

4) When is the acceleration negative for 0<= t <= 2pi
ANS: 0 < t < pi

5) Calculate the total distance travelled by the particle in the first 'pi' seconds. (Just show me this, thanks!)
ANS: 2√3 + pi/3

 

[qwerty44]

Re: A particle P is moving along the x-axis. Its position at time t seconds is given

A particle P is moving along the x-axis. Its position at time t seconds is given by:

x = 2sint - t , where t >= 0

[This is the order of the questions. I did 1-5 already, I just need 6]

1) Find an expression for the velocity of the particle.
ANS: V = 2cost - 1

2) In what direction is the particle moving at t = 0
ANS: right

3) Determine when the particle first comes to rest.
ANS: t = pi/3 seconds

4) When is the acceleration negative for 0<= t <= 2pi
ANS: 0 < t < pi

5) Calculate the total distance travelled by the particle in the first 'pi' seconds. (Just show me this, thanks!)
ANS: 2√3 + pi/3

Where is 6?

 

[Frostkruncher]

Re: A particle P is moving along the x-axis. Its position at time t seconds is given

ok, ill explain this question to you
5)Calculate the total distance travelled by the particle in the first 'pi' seconds.

Now, it means distance travelled, automatically know you have to refer to the integral of velocity.

x = 2sint - t
you have to either visualize it or sketch it
on the curve, you want to know how far it has travelled, thus, adding up all the displacement values in respect to the curvature.

The curve slights upwards from t=0 and hits a maximum at t=pi/3, call this distance D1
it goes back down, the SAME DISTANCE as D1 back to the x-axis, call this distance D2
it curves back down to t=pi, (the end of the domain), call this distance from x-axis to t=pi as Distance D3 (which will be negative)

Distance travelled = D1 + D2 + |D3|

D1=D2 (stated before)

Distance travelled = 2*D1 + |D3|
now, let f(t) = x
D1 = f(pi/3) = 2*sin(pi/3) - pi/3 = 2*(√3/2) - pi/3 = √3 - pi/3
D3 = f(pi) = 2sin(pi) - pi = - pi

Distance travelled = 2*(√3 - pi/3) + |-pi|
= 2√3 - 2pi/3 + pi
= 2√3 + pi/3

therefore, the distance this particle travels in pi seconds is 2√3 + pi/3