The position of a particle moving on the x-axis is given by `x=t^(3)+4t^(2)-2t+5` where `x` is in meter and `t` is in seconds
(i) the velocity of the particle at `t=4 s` is `78 m//s `
(ii) the acceleration of the particle at `t=4 s` is `32 m//s^(2)`
(iii) the average velocity during the interval `t=0` to `t=4 s `is `30 m//s`
(iv) the average acceleration during the interval `t=0` to `t=4 s` is `20 m//s^(2)`

Options

A. (i), (ii)
B. (ii), (iii)
C. (i), (ii), (iii)
D. All

Explanation

To solve the problem step by step, we will analyze the position function of the particle given by \( x(t) = t^3 + 4t^2 - 2t + 5 \) and find the required quantities: velocity, acceleration, average velocity, and average acceleration.

### Step 1: Find the Velocity Function
The velocity \( v(t) \) is the first derivative of the position function \( x(t) \) with respect to time \( t \).

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(t^3 + 4t^2 - 2t + 5) \]

Calculating the derivative:

\[ v(t) = 3t^2 + 8t - 2 \]

### Step 2: Calculate the Velocity at \( t = 4 \, s \)
Now, we substitute \( t = 4 \) into the velocity function:

\[ v(4) = 3(4^2) + 8(4) - 2 \]

Calculating each term:

\[ v(4) = 3(16) + 32 - 2 = 48 + 32 - 2 = 78 \, m/s \]

### Step 3: Find the Acceleration Function
The acceleration \( a(t) \) is the derivative of the velocity function \( v(t) \).

\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(3t^2 + 8t - 2) \]

Calculating the derivative:

\[ a(t) = 6t + 8 \]

### Step 4: Calculate the Acceleration at \( t = 4 \, s \)
Now, we substitute \( t = 4 \) into the acceleration function:

\[ a(4) = 6(4) + 8 \]

Calculating:

\[ a(4) = 24 + 8 = 32 \, m/s^2 \]

### Step 5: Calculate the Average Velocity from \( t = 0 \) to \( t = 4 \)
The average velocity \( \bar{v} \) over an interval is given by:

\[ \bar{v} = \frac{x(t_f) - x(t_i)}{t_f - t_i} \]

Where \( t_i = 0 \) and \( t_f = 4 \).

First, we calculate \( x(0) \):

\[ x(0) = (0)^3 + 4(0)^2 - 2(0) + 5 = 5 \]

Next, we calculate \( x(4) \):

\[ x(4) = (4)^3 + 4(4)^2 - 2(4) + 5 = 64 + 64 - 8 + 5 = 125 \]

Now, we can find the average velocity:

\[ \bar{v} = \frac{125 - 5}{4 - 0} = \frac{120}{4} = 30 \, m/s \]

### Step 6: Calculate the Average Acceleration from \( t = 0 \) to \( t = 4 \)
The average acceleration \( \bar{a} \) is given by:

\[ \bar{a} = \frac{a(t_f) - a(t_i)}{t_f - t_i} \]

Where \( a(t_i) = a(0) \) and \( a(t_f) = a(4) \).

Calculating \( a(0) \):

\[ a(0) = 6(0) + 8 = 8 \, m/s^2 \]

Now, we already found \( a(4) = 32 \, m/s^2 \).

Now, we can find the average acceleration:

\[ \bar{a} = \frac{32 - 8}{4 - 0} = \frac{24}{4} = 6 \, m/s^2 \]

### Summary of Results
(i) Velocity at \( t = 4 \, s \): \( 78 \, m/s \)
(ii) Acceleration at \( t = 4 \, s \): \( 32 \, m/s^2 \)
(iii) Average velocity from \( t = 0 \) to \( t = 4 \): \( 30 \, m/s \)
(iv) Average acceleration from \( t = 0 \) to \( t = 4 \): \( 6 \, m/s^2 \)