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A particle is moving in the xy plane with velocity and acceleration By taking the appropriate derivative, show that the magnitude of can be constant only if

A particle is moving in the xy plane with velocity  and acceleration By taking the appropriate derivative, show that the magnitude of  can be constant only if 

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
6 years ago
Given:
Velocity of particle,236-523_1.JPG
Acceleration of particle,236-453_2.JPG
The magnitude of velocity is given as:

236-162_3.JPG
Given that the magnitude of velocity is constant (say k), the above expression can be written as:
236-2111_4.JPG
Squaring both sides, we have
236-2247_5.JPG …… (1)
Now we will differentiate the above expression using the properties of differentiation given below:
236-1146_6.JPG
Using the property of differentiation above in equation (1), we have
236-1380_7.JPG …… (2)
Also, the acceleration is defined as the rate of change of velocity, therefore by differentiating the velocity vector\overrightarrow{v} we get acceleration vector\overrightarrow{a} as:
236-1503_8.JPG
However the given acceleration vector isa_{x}(t)\widehat{i} + a_{y}(t)\widehat{j} . Therefore on comparing the given acceleration vector with the equation above, we have
236-131_9.JPG
Substitute the above condition in equation (2) to get
236-689_10.JPG
Therefore, only when the conditionv_{x}(t)a_{x}(t)+v_{y}(t)a_{y}(t) = 0 is satisfied, the magnitude of the velocity vector \overrightarrow{v} will be constant.

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