A particle is moving with a velocity of `vec(v)=(3hat(i)+4that(j))m//s`. Find the ratio of rangential acceleration to that of total acceleration at `t=1sec`

Options

A. `(4)//(5)`
B. `(3)//(5)`
C. `(5)//(4)`
D. `(3)//(4)`

Explanation

To solve the problem of finding the ratio of tangential acceleration to total acceleration for a particle moving with a velocity given by \(\vec{v} = (3\hat{i} + 4t\hat{j}) \, \text{m/s}\) at \(t = 1 \, \text{s}\), we can follow these steps:

### Step 1: Determine the velocity at \(t = 1 \, \text{s}\)

Substituting \(t = 1\) into the velocity equation:

\[ \vec{v} = 3\hat{i} + 4(1)\hat{j} = 3\hat{i} + 4\hat{j} \, \text{m/s} \]

### Step 2: Calculate the magnitude of the velocity

The magnitude of the velocity \(|\vec{v}|\) is given by:

\[ |\vec{v}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]

### Step 3: Find the acceleration of the particle

To find the acceleration, we differentiate the velocity with respect to time:

\[ \vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(3\hat{i} + 4t\hat{j}) = 0\hat{i} + 4\hat{j} = 4\hat{j} \, \text{m/s}^2 \]

### Step 4: Identify tangential and radial components of acceleration

1. **Tangential Acceleration (\(a_t\))**: This is the component of acceleration in the direction of the velocity. Since the acceleration is purely in the \(\hat{j}\) direction and the velocity has both \(\hat{i}\) and \(\hat{j}\) components, we can find the tangential acceleration using the formula:

\[ a_t = \frac{d|\vec{v}|}{dt} \]

Since the speed is changing only in the \(\hat{j}\) direction, we can directly take the component of acceleration along the direction of velocity.

2. **Total Acceleration (\(a\))**: The total acceleration is the magnitude of the acceleration vector:

\[ |\vec{a}| = \sqrt{(0)^2 + (4)^2} = 4 \, \text{m/s}^2 \]

### Step 5: Calculate the ratio of tangential acceleration to total acceleration

To find the ratio of tangential acceleration to total acceleration, we can use the formula:

\[ \text{Ratio} = \frac{a_t}{|\vec{a}|} \]

Since the tangential acceleration is the component of acceleration in the direction of the velocity, we can find it using the dot product of the acceleration and the unit vector of velocity.

The unit vector of velocity \(\hat{v}\) is:

\[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{3\hat{i} + 4\hat{j}}{5} \]

Now, we can find the tangential acceleration:

\[ a_t = \vec{a} \cdot \hat{v} = (4\hat{j}) \cdot \left(\frac{3\hat{i} + 4\hat{j}}{5}\right) = \frac{4 \cdot 4}{5} = \frac{16}{5} \, \text{m/s}^2 \]

### Step 6: Calculate the ratio

Now we can calculate the ratio:

\[ \text{Ratio} = \frac{a_t}{|\vec{a}|} = \frac{\frac{16}{5}}{4} = \frac{16}{20} = \frac{4}{5} \]

### Final Answer

The ratio of tangential acceleration to total acceleration at \(t = 1 \, \text{s}\) is:

\[ \frac{4}{5} \]

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