# Calculus

Two particles move along the x -axis. For 0 is less than or equal to t is less than or equal to 6, the position of particle P at time t is given by p(t)=2cos((pi/4)t), while the position of particle R at time t is given by r(t)=t^3 -6t^2 +9t+3.

1. For 0 is less than or equal to t is less than or equal to 6, find all times t during which particle R is moving to the left. 2. for 0 is less than or equal to t is less than or equal to 6, find all times t during which the two particles travel in opposite directions. 3. Find the acceleration of particles P at time t=3. Is particle P speeding up, slowing down, or doing neither at time t=3? Explain your reasoning. 4. Show that during the interval (1,3), there must be at least one instant when the particle R must have a velocity of -2.

1. 👍 0
2. 👎 0
3. 👁 228
1. 1. p '(t) = -2(π/4) sin(π/4t) , which is the velocity
to move to the left, p '(t) < 0 or negative, so
sin (π/4t) must be positive.
the period of sin(π/4t) is 2π/(π/4) = 8
so for the first 4 seconds the curve is above the x-axis or it is positive
so for 0 ≤ t ≤ 6, the particle is moving to the left for the first 4 seconds, then to the right for the next 2 seconds.

ARGGGG , it asked for particle R and I found it for P, Oh well, maybe we can use the above later on

r ' (t) = 3t^2 - 12t + 9
= 3(t-1)(t-3)
zeros are t = 1 and t = 3
so r'(t) is positive for t < 1 or t > 3 and is negative for 1 < t < 3

make a sketch of both derivatives on the same grid
look at when both curves are on the same side of the x-axis ---> going same direction
when both curves are on opposite sides of the x-axis --> same direction

From 0 to 1, R is moving to the right,
from 1 to 3, R is moving to the left
from 3 to 6 , R is moving to the right again.

So from 0 to 1, they are moving in the opposite directions
From 1 to 3, they are moving in the same direction , (both v's are negative
from 3 to 4 , they are moving in opposites
from 4 to 6 , they are moving in same directions

3. r '' (t) = 6t - 12
r ''(3) = 18-12 which is positive, so r ' (t) or the velocity of R is increasing.

4. when is 3t^2 - 12t + 9 = -2
3t^2 - 12t + 11 = 0
t = (12 ± √12)/6 = appr 2.577 or 1.423

both t = 2.577 and t = 1.423 fall in the interval 1 ≤ t ≤ 3

YEAHHH

1. 👍 0
2. 👎 0
posted by Reiny

## Similar Questions

1. ### Calculus

Two particles move along the x -axis. For 0 is less than or equal to t is less than or equal to 6, the position of particle P at time t is given by p(t)=2cos((pi/4)t), while the position of particle R at time t is given by

2. ### physics

Three particles of equal masses travelling with velocities of 10m/s,20m/s and 30m/s respectively along x-axis at an angle of 30° to the direction of positive x-axis and Y-axis collide simultaneously and get sticked to each other

asked by Thousif on August 21, 2017
3. ### Science

Why is a gas able to flow? It's particles have melted and can move around. It's particles have high viscosity and can move around. It's particles have enough energy to move around. It's particles have enough pressure to move

asked by Sally on April 6, 2016
4. ### Science

Which of the foolowing occurs when a mass of air rises into the atmosphere? A) it becomes warmer because the particles move closer together B) it becomes cooler because the particles move closer together C) it becomes cooler

asked by Astrid on April 4, 2012
5. ### College Physics

Two particles are fixed to an x axis: particle 1 of charge q1 = 2.22 x 10-8 C at x = 15.6 cm and particle 2 of charge q2 = -4.33q1 at x = 66.7 cm. At what coordinate on the x axis is the electric field produced by the particles

asked by Stephanie22-11 on July 8, 2012
6. ### Physics

Two particles are fixed to an x axis: particle 1 of charge q1 = 1.73 ¡Ñ 10-8 C at x = 30.0 cm and particle 2 of charge q2 = -4.41q1 at x = 78.0 cm. At what coordinate on the x axis is the electric field produced by the particles

asked by Anonymous on October 20, 2015
7. ### calculus

Two particles move along the x-axis. For 0≤t≤6, the position of particle P is p(t)=2cos(t*π/4), while the position of particle R is r(t)=t^3-6t^2+9t+3. For 0≤t≤6 find all the times which the two particles travel in

asked by nico on November 25, 2012
8. ### Class 11th Physics

I have read about Rotatory Motion it says "It is that motion in hwich body, which is not a point mass body is moving such that all its constituent particles move simultaneously along concentric circles, whose centres lie on a

asked by Akashdeep on August 6, 2009
9. ### Chemistry

Why during melting, particles gain energy and vibrate more, until they are able to overcome the forces that hold them in their fixed position? Why during boiling, particles gain kinetic energy and move quickly until they can

asked by StuartKess on September 9, 2011
10. ### Chemistry

Which of these most accurately describes the motion of gas particles? 1.) particles making up gases are mostly fixed in their locations and have vibrational motion. 2.) particles making up gases have few intermolecular

asked by Kayla on February 19, 2018

More Similar Questions