The displacement is given by `x = 2t^(2) +t +5`, the acceleration at `t = 2s` is

Options

A. `4m//s^(2)`
B. `8m//s^(2)`
C. `10m//s^(2)`
D. `15m//s^(2)`

Answer

To find the acceleration at \( t = 2 \, \text{s} \) given the displacement function \( x = 2t^2 + t + 5 \), we will follow these steps:

### Step 1: Write down the displacement equation.
The displacement is given by:
\[ x(t) = 2t^2 + t + 5 \]

### Step 2: Find the first derivative of the displacement to get the velocity.
The velocity \( v(t) \) is the first derivative of the displacement \( x(t) \) with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(2t^2 + t + 5) \]

Calculating the derivative:
\[ v(t) = 4t + 1 \]

### Step 3: Find the second derivative of the displacement to get the acceleration.
The acceleration \( a(t) \) is the derivative of the velocity \( v(t) \):
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(4t + 1) \]

Calculating the derivative:
\[ a(t) = 4 \]

### Step 4: Evaluate the acceleration at \( t = 2 \, \text{s} \).
Since the acceleration is constant and equals \( 4 \, \text{m/s}^2 \), we find:
\[ a(2) = 4 \, \text{m/s}^2 \]

### Final Answer:
The acceleration at \( t = 2 \, \text{s} \) is:
\[ \boxed{4 \, \text{m/s}^2} \]

---