For any particle moving with some velocity `(vecv)` & acceleration `(veca)`, tangential acceleration & normal acceleration are defined as follows.
Tangential acceleration - The component of acceleration in the direction of velocity.
Normal acceleration - The component of acceleration in the direction perpendicular to velocity.
If at a given instant, velocity & acceleration of a particle are given by .
`vecc=4hati +3hatj`
`vaca=10hati+15hatj+20hatk`
Find the tangential acceleration of the particle at the given instant :-

Options

A. `17(4hati+3hatj)`
B. `(17)/(5)(4hati+3hatj)`
C. `17(4hati-3hatj)`
D. `(17)/(5)(4hati-3hatj)`

Answer

To find the tangential acceleration of the particle at the given instant, we will follow these steps:

### Step 1: Identify the given vectors
The velocity vector \( \vec{v} \) and acceleration vector \( \vec{a} \) are given as:
\[ \vec{v} = 4\hat{i} + 3\hat{j} \]
\[ \vec{a} = 10\hat{i} + 15\hat{j} + 20\hat{k} \]

### Step 2: Calculate the magnitude of the velocity vector
The magnitude of the velocity vector \( |\vec{v}| \) is calculated as:
\[ |\vec{v}| = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Calculate the dot product of acceleration and velocity
Next, we need to calculate the dot product \( \vec{a} \cdot \vec{v} \):
\[ \vec{a} \cdot \vec{v} = (10\hat{i} + 15\hat{j} + 20\hat{k}) \cdot (4\hat{i} + 3\hat{j}) \]
Calculating this gives:
\[ = 10 \cdot 4 + 15 \cdot 3 + 20 \cdot 0 = 40 + 45 + 0 = 85 \]

### Step 4: Calculate the tangential acceleration
The tangential acceleration \( a_t \) is given by the formula:
\[ a_t = \frac{\vec{a} \cdot \vec{v}}{|\vec{v}|} \]
Substituting the values we calculated:
\[ a_t = \frac{85}{5} = 17 \]

### Step 5: Find the direction of the tangential acceleration
The unit vector in the direction of velocity \( \hat{v} \) is:
\[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{4\hat{i} + 3\hat{j}}{5} = \frac{4}{5}\hat{i} + \frac{3}{5}\hat{j} \]

Thus, the tangential acceleration vector \( \vec{a_t} \) is:
\[ \vec{a_t} = a_t \cdot \hat{v} = 17 \left(\frac{4}{5}\hat{i} + \frac{3}{5}\hat{j}\right) = \frac{68}{5}\hat{i} + \frac{51}{5}\hat{j} \]

### Final Answer
The tangential acceleration of the particle at the given instant is:
\[ \vec{a_t} = \frac{68}{5}\hat{i} + \frac{51}{5}\hat{j} \]

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