If the displacement of a particle is givne by `s=((1)/(2)t^(2)+4sqrtt)m`. Find the velocity and acceleration at t = 4 seconds.

Answer

To solve the problem step by step, we will first find the velocity and then the acceleration of the particle at \( t = 4 \) seconds.

### Step 1: Write down the displacement equation
The displacement of the particle is given by:
\[ s = \frac{1}{2} t^2 + 4 \sqrt{t} \text{ meters} \]

### Step 2: Differentiate the displacement to find velocity
Velocity \( v \) is the rate of change of displacement with respect to time, which can be expressed as:
\[ v = \frac{ds}{dt} \]
Now, we differentiate \( s \) with respect to \( t \):
\[ v = \frac{d}{dt} \left( \frac{1}{2} t^2 + 4 \sqrt{t} \right) \]

Using the power rule for differentiation:
- The derivative of \( \frac{1}{2} t^2 \) is \( t \).
- The derivative of \( 4 \sqrt{t} \) (which is \( 4 t^{1/2} \)) is \( 4 \cdot \frac{1}{2} t^{-1/2} = \frac{2}{\sqrt{t}} \).

So, we have:
\[ v = t + \frac{2}{\sqrt{t}} \]

### Step 3: Calculate the velocity at \( t = 4 \) seconds
Now, we substitute \( t = 4 \) into the velocity equation:
\[ v(4) = 4 + \frac{2}{\sqrt{4}} = 4 + \frac{2}{2} = 4 + 1 = 5 \text{ m/s} \]

### Step 4: Differentiate the velocity to find acceleration
Acceleration \( a \) is the rate of change of velocity with respect to time, which can be expressed as:
\[ a = \frac{dv}{dt} \]
Now, we differentiate \( v \) with respect to \( t \):
\[ a = \frac{d}{dt} \left( t + \frac{2}{\sqrt{t}} \right) \]

Differentiating term by term:
- The derivative of \( t \) is \( 1 \).
- The derivative of \( \frac{2}{\sqrt{t}} \) (which is \( 2 t^{-1/2} \)) is \( -\frac{1}{2} \cdot 2 t^{-3/2} = -\frac{1}{t^{3/2}} \).

So, we have:
\[ a = 1 - \frac{1}{t^{3/2}} \]

### Step 5: Calculate the acceleration at \( t = 4 \) seconds
Now, we substitute \( t = 4 \) into the acceleration equation:
\[ a(4) = 1 - \frac{1}{4^{3/2}} = 1 - \frac{1}{8} = 1 - 0.125 = 0.875 \text{ m/s}^2 \]

### Final Results
- The velocity at \( t = 4 \) seconds is \( 5 \) m/s.
- The acceleration at \( t = 4 \) seconds is \( 0.875 \) m/s².