M2

imasillynarb

Two particles A and B move in the plane of cartesian coordinate axes Ox, Oy. At time t seconds the position vectors of A and B, reffered to the origin O, are (t^2i + 4tj)m and (2ti + (t+1)j)m respectively.

a) prove that the particles never collide

How do I do this? Ive forgotten this from M1? I thought you just equated the coefficients, but when I do this I get t = 2 and a 1/3 meaning they do collide no?

c) Find the value of t when A and B have parallel velocities, and find the distance between A and B at this instant.

Cant remember how to do this either..

well, the only way for them to collide is if they have the same position vector at a value of t.

Reply 2

a) yes, you equate coefficients, and try (and fail) to solve them simultaneously

c) parallel velocities means that vA = k vB, where k is a scalar. so you equate coefficients, multiplying one of the velocities by k, and solving simultaneously to get t

Reply 3

OP

IntegralAnomaly

well, the only way for them to collide is if they have the same position vector at a value of t.

So equating the coefficients is right? If they never collide then why do I get 2 values of t..

Reply 4

OP

elpaw

a) yes, you equate coefficients, and try (and fail) to solve them simultaneously

c) parallel velocities means that vA = k vB, where k is a scalar. so you equate coefficients, multiplying one of the velocities by k, and solving simultaneously to get t

So the particles DO collide?

Reply 5

imasillynarb

So equating the coefficients is right? If they never collide then why do I get 2 values of t..

thats the point, if they did collide, you would get 2 values of t that are the same

Reply 6

OP

elpaw

thats the point, if they did collide, you would get 2 values of t that are the same

Im confused. If they collide, then when you equate the coeffs you will get a value of t? So if they dont collide, the t's must cancel ?

Reply 7

OP

Actually, I understand now, but I get t = 2 and t = 1/3, clearly they arent the same, therefore they collide?

Reply 8

imasillynarb

So equating the coefficients is right? If they never collide then why do I get 2 values of t..

For the second bit u differentiate wrt to time,and as elpaw said if they are parallel then a scalar times the velocity of one is equal to the other

Reply 9

OP

IntegralAnomaly

For the second bit u differentiate wrt to time,and as elpaw said if they are parallel then a scalar times the velocity of one is equal to the other

OK, if so, v = 4 for A and v = 6 - 2t for B

4 = 6k - 2kt

t = (4 - 6k)/2k

Which means I have t and k, how can I find the time ?

Reply 10

OP

Whhops, ignore that, Im using the numbers for the next question here...

Reply 11

OP

Ahar, wait, with part a)

If I equate the coefficients and get 2 values of t the same then they collide, but if they are different, they dont? Is that what you were saying?

Reply 12

imasillynarb

OK, if so, v = 4 for A and v = 6 - 2t for B

4 = 6k - 2kt

t = (4 - 6k)/2k

Which means I have t and k, how can I find the time ?

for a dr/dt=2ti+4j , and for b dr/dt=2i+j

Reply 13

imasillynarb

Actually, I understand now, but I get t = 2 and t = 1/3, clearly they arent the same, therefore they collide?

no.

first you equate i coefficients, getting t=2.

if you plug t=2 into the j coefficients, you get 8 for A and 3 for B. this means at time t=2, they have the coordinates (2,8) and (2,3), so they do not collide

alternitavely, if you start by equating j coefficients, you get t=1/3. if you plug it into the i coefficients, you get 1/9 for a and 2/3 for b. thus at time t=1/3 they are at coords (1/9, 1/3) and (2/3, 1/3), so they do not collide.

the only way for them to collide is if the equation for the i coefficients and the equation for the j coefficients to give the same value for t when solved.

Reply 14

imasillynarb

Ahar, wait, with part a)

If I equate the coefficients and get 2 values of t the same then they collide, but if they are different, they dont? Is that what you were saying?

yes, exactly

Reply 15

Juwel

18

imasillynarb

Ahar, wait, with part a)

If I equate the coefficients and get 2 values of t the same then they collide, but if they are different, they dont? Is that what you were saying?

Point being that they have the same position (vector) at different times (t).

Reply 16