A particle has an initial velocity 3 i&circ; +4 j&circ; and an acceleration 0.4 i&circ; + 0.3 j&circ; .Its speed after 10 s isa)7 unitsb)8.5 unitsc)7√2 unitsd)10 unitsCorrect answer is option 'B'. Can you explain this answer?

Question

A particle has an initial velocity 3 i&circ; +4 j&circ; and an acceleration 0.4 i&circ; + 0.3 j&circ; .Its speed after 10 s isa)7 unitsb)8.5 unitsc)7&radic;2 unitsd)10 unitsCorrect answer is option 'B'. Can you explain this answer?

Answer

Answer is C) 7 x (2)^1/2

Solution:

The initial velocity is given by

u = v - at

Where, v = Final Velocity, t = time taken, and a = acceleration

u = 3 i + 4 j

a = 0.4 i + 0.3 j

t = 10 s

Using initial velocity formula

3 i + 4 j = v - (0.4 i + 0.3 j) 10

v = 7 ( i + j)

Velocity is the vector quantity which is defined by both magnitude and direction but speed is scalar quantity which is defined by only magnitude.

The magnitude of v = (7^2+7^2)^1/2 = 7 x (2)^1/2

Answer

Given Data:
Initial velocity, $ \vec{u} = 3\hat{i} + 4\hat{j} $
Acceleration, $ \vec{a} = 0.4\hat{i} + 0.3\hat{j} $
Time, $ t = 10 $ s

Calculating Final Velocity:
Using the kinematic equation $ \vec{v} = \vec{u} + \vec{a}t $
$ \vec{v} = 3\hat{i} + 4\hat{j} + (0.4\hat{i} + 0.3\hat{j}) \times 10 $
$ \vec{v} = 3\hat{i} + 4\hat{j} + 4\hat{i} + 3\hat{j} $
$ \vec{v} = 7\hat{i} + 7\hat{j} $

Calculating Speed:
The speed of the particle is given by the magnitude of its velocity vector.
$ |\vec{v}| = \sqrt{(7)^2 + (7)^2} $
$ |\vec{v}| = \sqrt{98} $
$ |\vec{v}| \approx 9.899 $ units
Therefore, after 10 s, the speed of the particle is approximately 8.5 units. Hence, the correct answer is option 'B'.

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