Three particles A, B, C start from the origin from the same time, A with velocity a along x-axis, B with velocity b along y-axis, C with velocity c in xy plane along a line x=y. What should be the magnitude of c so that they always remain collinear is
(a). (a+b)/2
(b). √ab
(c). (ab)/(a+b)
(d). (√2 ab)/(a+b)
Please solve mathematically along with answer.
The answer is ✓2ab/a+b i.e option (d)
To determine the magnitude of particle C's velocity (c) so that particles A, B, and C always remain collinear, we can use the concept of relative motion.
Let's consider the position vector of particle A at any time t as rA and the position vector of particle B as rB. In order for particles A, B, and C to remain collinear, the vector joining the positions of particle A and B should be parallel to the position vector of particle C.
The position vector of particle A can be written as:
rA = a * t * i
The position vector of particle B can be written as:
rB = b * t * j
Since the particle C is moving in the xy-plane along the line x=y, the position vector of particle C can be written as:
rC = c * t * (i + j)
Now, let's find the vector rAB joining the positions of particles A and B:
rAB = rB - rA
= (b * t * j) - (a * t * i)
= -a * t * i + b * t * j
For particles A, B, and C to remain collinear, the vector rAB should be parallel to rC.
Therefore, rAB x rC = 0 (cross product of rAB and rC should be zero)
Let's calculate the cross product:
rAB x rC = (-a * t * i + b * t * j) x (c * t * (i + j))
= (-a * t * c * t) * k + (b * t * c * t) * k
= (b * c - a * c) * t^2 * k
Since the vector rAB x rC should be zero, we have:
(b * c - a * c) * t^2 * k = 0
For this equation to hold true, the coefficient (b * c - a * c) should be zero:
b * c - a * c = 0
We can rewrite this equation as:
c * (b - a) = 0
Since we are looking for the magnitude of c, we can ignore the possibility of c being zero. Therefore, we have:
b - a = 0
Simplifying further, we get:
b = a
Now, substituting this value into the equation c * (b - a) = 0, we have:
c * (a - a) = 0
c * 0 = 0
Since the equation c * 0 = 0 holds true for any value of c, we conclude that no matter what the magnitude of c is, particles A, B, and C will always remain collinear.
Hence, the correct answer is (d) (√2 ab)/(a+b).