A particle is moving along a circular path ofradius R in such a way that at any instant magnitude of radial acceleration & tangential acceleration are equal. 1f at t = 0 velocity of particle is `V_(0)`. Find the speed of the particle after time `t=(R )//(2V_(0))`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the relationship between radial and tangential acceleration
Given that the magnitude of radial acceleration \( a_r \) is equal to the tangential acceleration \( a_t \), we can express this as:
\[ a_t = a_r \]
The tangential acceleration \( a_t \) can be expressed as the derivative of velocity with respect to time:
\[ a_t = \frac{dV}{dt} \]
The radial acceleration \( a_r \) is given by:
\[ a_r = \frac{V^2}{R} \]
where \( V \) is the velocity of the particle and \( R \) is the radius of the circular path.

### Step 2: Set up the equation
Since \( a_t = a_r \), we can equate the two expressions:
\[ \frac{dV}{dt} = \frac{V^2}{R} \]

### Step 3: Rearrange the equation for integration
Rearranging the equation gives:
\[ \frac{dV}{V^2} = \frac{dt}{R} \]

### Step 4: Integrate both sides
We will integrate both sides. The left side will be integrated with respect to \( V \) from \( V_0 \) to \( V \) (the final speed we want to find), and the right side will be integrated with respect to \( t \) from \( 0 \) to \( \frac{R}{2V_0} \):
\[ \int_{V_0}^{V} \frac{dV}{V^2} = \int_{0}^{\frac{R}{2V_0}} \frac{dt}{R} \]

### Step 5: Solve the integrals
The integral on the left side is:
\[ -\frac{1}{V} \bigg|_{V_0}^{V} = -\frac{1}{V} + \frac{1}{V_0} \]
The integral on the right side is:
\[ \frac{1}{R} t \bigg|_{0}^{\frac{R}{2V_0}} = \frac{1}{R} \cdot \frac{R}{2V_0} = \frac{1}{2V_0} \]

### Step 6: Set the equations equal
Now we can set the two results equal to each other:
\[ -\frac{1}{V} + \frac{1}{V_0} = \frac{1}{2V_0} \]

### Step 7: Solve for \( V \)
Rearranging gives:
\[ \frac{1}{V} = \frac{1}{V_0} - \frac{1}{2V_0} = \frac{1}{2V_0} \]
Taking the reciprocal gives:
\[ V = 2V_0 \]

### Final Answer
Thus, the speed of the particle after time \( t = \frac{R}{2V_0} \) is:
\[ V = 2V_0 \]