The x and y coordinates of the particle at any time are `x=5t-2t^(2) and y=10t` respectively, where x and y are in meters and t in seconds. The acceleration of the particle at t=2s is:

Options

A. 0
B. `5m//s^(2)`
C. `-4m//s^(2)`
D. `-8m//s^(2)`

Explanation

To find the acceleration of the particle at \( t = 2 \) seconds, we will follow these steps:

### Step 1: Identify the equations of motion
The position of the particle is given by:
- \( x(t) = 5t - 2t^2 \)
- \( y(t) = 10t \)

### Step 2: Find the velocity in the x-direction
To find the acceleration, we first need to find the velocity. The velocity in the x-direction is the derivative of the x-position with respect to time:
\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(5t - 2t^2) \]
Calculating the derivative:
\[ v_x = 5 - 4t \]

### Step 3: Find the acceleration in the x-direction
The acceleration in the x-direction is the derivative of the velocity:
\[ a_x = \frac{dv_x}{dt} = \frac{d}{dt}(5 - 4t) \]
Calculating the derivative:
\[ a_x = -4 \, \text{m/s}^2 \]

### Step 4: Find the velocity in the y-direction
Now, we will find the velocity in the y-direction by differentiating the y-position:
\[ v_y = \frac{dy}{dt} = \frac{d}{dt}(10t) \]
Calculating the derivative:
\[ v_y = 10 \, \text{m/s} \]

### Step 5: Find the acceleration in the y-direction
The acceleration in the y-direction is the derivative of the velocity:
\[ a_y = \frac{dv_y}{dt} = \frac{d}{dt}(10) \]
Since the derivative of a constant is zero:
\[ a_y = 0 \, \text{m/s}^2 \]

### Step 6: Calculate the total acceleration
The total acceleration of the particle can be found using the components of acceleration in the x and y directions:
\[ \text{Total acceleration} = \sqrt{a_x^2 + a_y^2} \]
Substituting the values:
\[ \text{Total acceleration} = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4 \, \text{m/s}^2 \]

However, since the acceleration in the x-direction is negative, we can say:
\[ \text{Total acceleration} = -4 \, \text{m/s}^2 \, \text{(in the x-direction)} \]

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