# 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

7 years ago
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 we get acceleration vector as:

However the given acceleration vector is . 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 is satisfied, the magnitude of the velocity vector  will be constant.