If `vecu = 2hati - 2hatj + 3hatk` and the final velocity is `vecv = 2hati - 4hatj + 5hatk` and it is covered in a time of 10 sec, find the acceleration vector.

Options

A. `(3hati - 2hatj + 2hatk)/10`
B. `(-3hati + hatj + 2hatk)/10`
C. `(-3hati - 2hatj + 2hatk)/10`
D. `(-hati + hatk)/5`

Answer

To find the acceleration vector given the initial and final velocity vectors, we can follow these steps:

### Step 1: Identify the given vectors
We are given:
- Initial velocity vector, \( \vec{u} = 2\hat{i} - 2\hat{j} + 3\hat{k} \)
- Final velocity vector, \( \vec{v} = 2\hat{i} - 4\hat{j} + 5\hat{k} \)
- Time interval, \( \Delta t = 10 \, \text{s} \)

### Step 2: Calculate the change in velocity
The change in velocity \( \Delta \vec{v} \) is given by:
\[ \Delta \vec{v} = \vec{v} - \vec{u} \]
Substituting the values:
\[ \Delta \vec{v} = (2\hat{i} - 4\hat{j} + 5\hat{k}) - (2\hat{i} - 2\hat{j} + 3\hat{k}) \]

### Step 3: Perform the subtraction
Now, we subtract the vectors component-wise:
- For the \( \hat{i} \) component: \( 2 - 2 = 0 \)
- For the \( \hat{j} \) component: \( -4 - (-2) = -4 + 2 = -2 \)
- For the \( \hat{k} \) component: \( 5 - 3 = 2 \)

Thus, we have:
\[ \Delta \vec{v} = 0\hat{i} - 2\hat{j} + 2\hat{k} \]

### Step 4: Calculate the acceleration vector
The acceleration \( \vec{a} \) is given by the formula:
\[ \vec{a} = \frac{\Delta \vec{v}}{\Delta t} \]
Substituting the values:
\[ \vec{a} = \frac{0\hat{i} - 2\hat{j} + 2\hat{k}}{10} \]

### Step 5: Simplify the acceleration vector
Now, we simplify the vector:
\[ \vec{a} = 0\hat{i} - \frac{2}{10}\hat{j} + \frac{2}{10}\hat{k} = 0\hat{i} - \frac{1}{5}\hat{j} + \frac{1}{5}\hat{k} \]

Thus, the acceleration vector can be expressed as:
\[ \vec{a} = -\frac{1}{5}\hat{j} + \frac{1}{5}\hat{k} \]

### Final Answer
The acceleration vector is:
\[ \vec{a} = -\hat{j} + \hat{k} \, \text{(divided by 5)} \]

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