Two particles start from rest from the same point and move along the same straight path; the first starts with a uniform velocity of 20 m/minute and the second with a uniform acceleration of 5 m/ min^2. Before they meet again, when will their distance be maximum?

Category: QuestionsTwo particles start from rest from the same point and move along the same straight path; the first starts with a uniform velocity of 20 m/minute and the second with a uniform acceleration of 5 m/ min^2. Before they meet again, when will their distance be maximum?
Editor">Editor Staff asked 11 months ago

Two particles start from rest from the same point and move along the same straight path; the first starts with a uniform velocity of 20 m/minute and the second with a uniform acceleration of 5 m/ min^2. Before they meet again, when will their distance be maximum?
 
(a) At t = 8
 
(b) At t = 6
 
(c) At t = 4
 
(d) At t = 2
 
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Question is taken from Calculus Application in chapter Application of Calculus of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

The correct answer is (c) At t = 4
 
The explanation: Suppose, the two particle starts from rest at and move along the straight path OA.
 
Further assume that the distance between the particle is maximum after t minutes from start (before they meet again).
 
If we be the position of the particle after 30 minutes from the start which moves with uniform acceleration 5 m/min^2 and C that of the particle moving with uniform velocity 20 m/min, then we shall have,
 
OB = 1/2(5t^2) and OC = 20t
 
If the distance between the particle after t minutes from start be x m, then,
 
x = BC = OC – OB = 20t – (5/2)t^2
 
Now, dx/dt = 20 – 5t and d^2x/dt^2 = -5
 
For maximum or minimum values of x, we have,
 
dx/dt = 0
 
Or 20 – 5t = 0
 
Or t = 4
 
And [d^2y/dx^2] = -5 < 0
 
Thus, x is maximum at t = 4.