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)`

Answer

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

### Step 1: Find the expressions for velocity
The position coordinates of the particle are given as:
\[ x(t) = 5t - 2t^2 \]
\[ y(t) = 10t \]

To find the velocity, we need to differentiate these position equations with respect to time \( t \).

**Velocity in the x-direction:**
\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(5t - 2t^2) = 5 - 4t \]

**Velocity in the y-direction:**
\[ v_y = \frac{dy}{dt} = \frac{d}{dt}(10t) = 10 \]

### Step 2: Find the expressions for acceleration
Next, we differentiate the velocity equations to find the acceleration.

**Acceleration in the x-direction:**
\[ a_x = \frac{d^2x}{dt^2} = \frac{d}{dt}(5 - 4t) = -4 \]

**Acceleration in the y-direction:**
\[ a_y = \frac{d^2y}{dt^2} = \frac{d}{dt}(10) = 0 \]

### Step 3: Calculate the total acceleration
The total acceleration of the particle can be found using the components of acceleration in both x and y directions.

The total acceleration \( \mathbf{a} \) is given by:
\[ \mathbf{a} = (a_x, a_y) = (-4, 0) \]

### Step 4: Evaluate at \( t = 2 \) seconds
Since the acceleration components are constant, we can directly state that at \( t = 2 \) seconds:
\[ a_x = -4 \, \text{m/s}^2 \quad \text{and} \quad a_y = 0 \, \text{m/s}^2 \]

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

### Final Answer
The correct option is option 3: \(-4 \, \text{m/s}^2\).

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