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The position vector r of a moving particle at time t after the start of the motion is given by r = 5(1 + 4t)i + 5(19 + 2t−t2)j. Find the initial velocity of the particle. At time t = T, the particle is moving at right angles to its initial direction of motion. Find the value of T and the distance of the particle from its initial position at this time.

 The position vector r of a moving particle at time t after the start of the motion is given by r = 5(1 + 4t)i + 5(19 + 2t−t2)j. Find the initial velocity of the particle. At time t = T, the particle is moving at right angles to its initial direction of motion. Find the value of T and the distance of the particle from its initial position at this time. 

Grade:12th pass

2 Answers

Saurabh Koranglekar
askIITians Faculty 10335 Points
3 years ago
Dear student

To find the initial velocity put t =0
after differentiating r wrt t

Now At time T
put t=T
after differentiating r wrt t

also (Initial velo.)dot product(Velo at t=T) = 0 now T can be obtained

Regards
Bright Paintsil
11 Points
2 years ago
|Finding initial velocity,
V=dr/dt = 4i + (10-10t)j
But the initial velocity, t=0
V= 4i + [10-10(0)]j
V= 4i +10j
Now at time t=T
V= 4i + [10-10(T)]j
V= 4i + (10-10T)j
Moving at right angles to it initial direction 
V(0) . V(T) =0
(4i + 10j).[4i + (10-10T)j]
16+100-100T=0
100T=116
T=116/100
T=1.16
Now for the distance at t=T
We put T=1.16 into vector r and the find the magnitude 
r=5[1+4(1.16)]i +5[19 +2(1.16)-(1.16)(1.16)]j
|r|=|28.2*28.2+99.872*99.872|
|r|=103.7770 to 4d.p

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