Two particles A and B are initially at distance d apart. They starts m...
Solution:
Let t be the time taken for the particles to meet.
At time t, the distance travelled by A = vt.
At time t, the distance travelled by B = ut + d (since B is moving along a fixed straight line perpendicular to initial direction of motion of A).
Since the particles meet, the distance travelled by A = the distance travelled by B.
Therefore, vt = ut + d.
Solving for t, we get t = d/(v-u).
Therefore, the particles will meet after time d/(v-u).
Substituting v/(v-u) for t in the expression for distance travelled by A, we get the distance travelled by A before the particles meet.
Distance travelled by A = v * (d/(v-u)) = vd/(v-u).
Substituting u/(v-u) for t in the expression for distance travelled by B, we get the distance travelled by B before the particles meet.
Distance travelled by B = u * (d/(v-u)) + d = ud/(v-u) + d.
Since u^2 < v^2,="" we="" have="" v^2="" -="" u^2="" /> 0.
Therefore, (v-u)(v+u) > 0.
Dividing both sides of the equation vt = ut + d by (v-u), we get d/(v-u) = t - d/(v+u).
Substituting this in the expression for distance travelled by B, we get
Distance travelled by B = ud/(v-u) + d = ut + d = (v+u)t - d/(v+u) = (v+u)d/(v+u) - d/(v+u) = vd/(v+u).
Therefore, the particles will meet after travelling a total distance of vd/(v+u).
Subtracting the distance travelled by A before the particles meet from the total distance travelled by the particles before they meet, we get the distance travelled by B before the particles meet.
Distance travelled by B before the particles meet = vd/(v+u) - vd/(v-u) = 2vd/(v^2-u^2).
Therefore, the particles will meet after travelling a distance of 2vd/(v^2-u^2).
Dividing the distance travelled by B before the particles meet by the time taken for the particles to meet, we get the velocity of B.
Velocity of B = (2vd/(v^2-u^2))/(d/(v-u)) = 2u/(v+u).
Therefore, the particles will meet after time d/(v^2-u^2) and the answer is A.
Two particles A and B are initially at distance d apart. They starts m...
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