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

Question

Aditi Chauhan · Answer

Given:
Velocity of particle,

Acceleration of particle,

The magnitude of velocity is given as:

Given that the magnitude of velocity is constant (say k), the above expression can be written as:

Squaring both sides, we have

…… (1)
Now we will differentiate the above expression using the properties of differentiation given below:

Using the property of differentiation above in equation (1), we have

…… (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:

However the given acceleration vector is $a_{x}(t)\widehat{i} + a_{y}(t)\widehat{j}$ . Therefore on comparing the given acceleration vector with the equation above, we have

Substitute the above condition in equation (2) to get

Therefore, only when the condition $v_{x}(t)a_{x}(t)+v_{y}(t)a_{y}(t) = 0$ is satisfied, the magnitude of the velocity vector $\overrightarrow{v}$ will be constant.

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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

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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

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