Two particles A and B have velocities 3^i and v^j respectively (in m/s).
(a) Find rBjA; the position of B relative to A for all t given that
rBjA(t = 0) = 9^i + 6^j (in metres).
(b) Find the value of v such that A and B collide.
(c) If v = 1 m/s, �nd the time and distance when A and B are closest together.
To find the position of particle B relative to particle A at a given time t, we can use the formula:
rBjA(t) = rBjA(t=0) + vB(t) - vA(t)
where rBjA(t=0) is the initial position of B relative to A, vB(t) is the velocity of B at time t, and vA(t) is the velocity of A at time t.
(a) Let's find rBjA for all t using the given information:
Given:
rBjA(t = 0) = 9^i + 6^j
We are given that the velocity of particle A is 3^i, and the velocity of particle
B is v^j. Therefore, the velocities of A and B are constant, and we can write vA(t) = 3^i and vB(t) = v^j.
Using the formula, we substitute the values:
rBjA(t) = (9^i + 6^j) + v^j - 3^i
(b) To find the value of v at which particles A and B collide, we need to find the time when their positions are equal. This means rBjA(t) = 0.
Substituting rBjA(t) = 0 into the equation from part (a), we get:
(9^i + 6^j) + v^j - 3^i = 0
Simplifying this equation, we obtain:
(9^i - 3^i) + (v^j + 6^j) = 0
Since particles A and B collide, their positions are equal at the time of collision. Thus, (9^i - 3^i) and (v^j + 6^j) must be equal.
Therefore, v^j + 6^j = 9^i - 3^i
By equating the corresponding components, we have:
v = 9 - 3 = 6 m/s
Therefore, the value of v at which particles A and B collide is 6 m/s.
(c) To find the time and distance when particles A and B are closest together, we need to find the minimum distance between their positions.
Given that v = 1 m/s, we can substitute this value into the equation from part (a) to find rBjA(t). Then we can calculate the magnitude of rBjA(t) to find the distance between A and B.
Using v = 1 m/s in the equation from part (a):
rBjA(t) = (9^i + 6^j) + (v^j - 3^i) = (9^i + 6^j) + (1^j - 3^i)
Now we can find the magnitude of rBjA(t):
| rBjA(t) | = sqrt((9^i + 6^j + 1^j - 3^i)^2)
Then, we can find the time when particles A and B are closest together by minimizing | rBjA(t) | with respect to time t.
Once we find the minimum value of | rBjA(t) |, we can use that value to calculate the distance between particles A and B.