The position vector of an object moving in X-Z plane is `vec(r)=v_(0)that(i)+a_(0)e^(b_(0)t)hat(k)`.
Find its (i) velocity `(vec(v)=(dvec(r))/(dt))` (ii) speed `(|vec(v)|)` (iii) Acceleration `((dvec(v))/(dt))` as a function of time.

Explanation

To solve the problem step by step, we will find the velocity, speed, and acceleration of the object moving in the X-Z plane given the position vector.

### Given:
The position vector of the object is:
\[ \vec{r} = v_0 t \hat{i} + a_0 e^{b_0 t} \hat{k} \]

### (i) Finding Velocity \(\vec{v} = \frac{d\vec{r}}{dt}\)

1. **Differentiate the position vector with respect to time \(t\)**:
- The position vector \(\vec{r}\) consists of two components:
- The \(x\)-component: \(v_0 t \hat{i}\)
- The \(z\)-component: \(a_0 e^{b_0 t} \hat{k}\)

2. **Differentiate each component**:
- For the \(x\)-component:
\[ \frac{d}{dt}(v_0 t) = v_0 \]
- For the \(z\)-component:
\[ \frac{d}{dt}(a_0 e^{b_0 t}) = a_0 b_0 e^{b_0 t} \]

3. **Combine the results**:
- Thus, the velocity vector is:
\[ \vec{v} = v_0 \hat{i} + a_0 b_0 e^{b_0 t} \hat{k} \]

### (ii) Finding Speed \(|\vec{v}|\)

1. **Calculate the magnitude of the velocity vector**:
- The speed is the magnitude of the velocity vector:
\[ |\vec{v}| = \sqrt{(v_0)^2 + (a_0 b_0 e^{b_0 t})^2} \]

2. **Simplify the expression**:
- Therefore, the speed is:
\[ |\vec{v}| = \sqrt{v_0^2 + (a_0 b_0)^2 e^{2b_0 t}} \]

### (iii) Finding Acceleration \(\vec{a} = \frac{d\vec{v}}{dt}\)

1. **Differentiate the velocity vector with respect to time \(t\)**:
- The velocity vector is:
\[ \vec{v} = v_0 \hat{i} + a_0 b_0 e^{b_0 t} \hat{k} \]

2. **Differentiate each component**:
- For the \(x\)-component:
\[ \frac{d}{dt}(v_0) = 0 \]
- For the \(z\)-component:
\[ \frac{d}{dt}(a_0 b_0 e^{b_0 t}) = a_0 b_0^2 e^{b_0 t} \]

3. **Combine the results**:
- Thus, the acceleration vector is:
\[ \vec{a} = 0 \hat{i} + a_0 b_0^2 e^{b_0 t} \hat{k} \]

### Final Results:
- **Velocity**:
\[ \vec{v} = v_0 \hat{i} + a_0 b_0 e^{b_0 t} \hat{k} \]
- **Speed**:
\[ |\vec{v}| = \sqrt{v_0^2 + (a_0 b_0)^2 e^{2b_0 t}} \]
- **Acceleration**:
\[ \vec{a} = 0 \hat{i} + a_0 b_0^2 e^{b_0 t} \hat{k} \]