What the random... (1 Viewer)

[xwrathbringerx]

Two particles (A and B) move on a straight line and their position (x metres) from the origin O at time t hours is givne by x = sint and sin2t respectively. Write down an interval of time in which both particles are travelling towards the origin with positive velocity.

Could someone please explain to me how on earth to do this question?<!-- google_ad_section_end -->

 

[lucymowat]

find out when they have negative position but positive velocity... right?

 

[Xcelz]

poster above is correct

so when x < 0
and dx/dt is positive.

 

[xwrathbringerx]

What exactly is the topic 4 this type of question?

 

[kaz1]

What exactly is the topic 4 this type of question?

Simple Harmonic Motion

 

[Drongoski]

Two particles (A and B) move on a straight line and their position (x metres) from the origin O at time t hours is givne by x = sint and sin2t respectively. Write down an interval of time in which both particles are travelling towards the origin with positive velocity.

Could someone please explain to me how on earth to do this question?<!-- google_ad_section_end -->

Both are moving in simple harmonic motion, oscillating between x = -1 and x = 1.

For A: dx/dt = cos t and x is < 0 and dx/dt > 0 for 3pi/2 < t < 2pi

For B: dx/dt = 2cos2t; x <0 and dx/dt >0 for (amongst various other intervals):

2pi + 3pi/2 < 2t < 2pi + 2pi ==> 7pi/4 < t < 2pi

.: for 7pi/4 < t < 2pi: both A & B are heading towards the origin with positive velocity.

 

[Drongoski]

A not very fluent explanation

0 < t < pi/2: x is +ve A's velocity is +ve: so A is heading from origin to the right to x = 1

0 < 2t < pi/2: x is +ve B's velocity is +ve: B is heading from origin to the right towards x = 1

So the common time interval: 0 < t < pi/4 is not a valid answer.

We need t for A and 2t for B to be in the 4th quadrant: where x is -ve and velocity is +ve.

For A: 3pi/2 < t < 2pi satisfies the requirements.
For B: 3pi/2 < 2t < 2pi (i.e. 3pi/4 < t < pi) satisfies this requirements for B but is not "simultaneous" with A above.
For B: 7pi/2 < 2t < 4pi i.e 7pi/4 < t < 2pi: B satisfies the said requirements and is part of A's time interval [3pi/2, 2pi]. So 7pi/4 < t < 2pi is the time interval when both A and B are heading towards the origin from the left with positive velocity. There are an infinite number of other simultaneous time intervals that satisfy these requirement.