A particle travels according to the equation `y=x-(x^(2)/(2))`. Find the maximum height it acheives (x and y are in metre)

Options

A. 1 m
B. 1/3 m
C. 1/2 m
D. 1/4 m

Answer

To find the maximum height achieved by the particle traveling according to the equation \( y = x - \frac{x^2}{2} \), we can follow these steps:

### Step 1: Differentiate the equation
We start by differentiating \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 1 - x \]

### Step 2: Set the derivative to zero
To find the critical points where the maximum or minimum height occurs, we set the derivative equal to zero:
\[ 1 - x = 0 \]
Solving for \( x \) gives:
\[ x = 1 \text{ meter} \]

### Step 3: Check if it's a maximum or minimum
To determine whether this critical point is a maximum or minimum, we need to find the second derivative:
\[ \frac{d^2y}{dx^2} = -1 \]
Since the second derivative is negative (\(-1 < 0\)), this indicates that the function has a maximum at \( x = 1 \) meter.

### Step 4: Calculate the maximum height
Now, we substitute \( x = 1 \) back into the original equation to find the maximum height \( y \):
\[ y = 1 - \frac{(1)^2}{2} = 1 - \frac{1}{2} = \frac{1}{2} \text{ meters} \]

### Conclusion
The maximum height achieved by the particle is \( \frac{1}{2} \) meters.