A projectile is given an initial velocity of `(hat(i)+2hat(j))` The Cartesian equation of its path is `(g = 10 ms^(-1))` ( Here , `hati` is the unit vector along horizontal and `hatj` is unit vector vertically upwards)

Options

A. (a)`y = 2x - 5x^2`
B. (b)`y = x - 5x^2`
C. (c)`4y=2x-5x^2`
D. (d)`y=2x-25x^2`

Answer

To find the Cartesian equation of the projectile's path given the initial velocity vector \(\hat{i} + 2\hat{j}\) and the acceleration due to gravity \(g = 10 \, \text{m/s}^2\), we can follow these steps:

### Step 1: Identify the components of the initial velocity
The initial velocity vector can be broken down into its components:
- Horizontal component \(u_x = 1 \, \text{m/s}\) (coefficient of \(\hat{i}\))
- Vertical component \(u_y = 2 \, \text{m/s}\) (coefficient of \(\hat{j}\))

### Step 2: Use the equations of motion
The equations of motion for a projectile can be used to find the relationship between \(x\) and \(y\):
- The horizontal motion is given by:
\[ x = u_x t \]
- The vertical motion is given by:
\[ y = u_y t - \frac{1}{2} g t^2 \]

### Step 3: Express time \(t\) in terms of \(x\)
From the horizontal motion equation:
\[ t = \frac{x}{u_x} = \frac{x}{1} = x \]

### Step 4: Substitute \(t\) into the vertical motion equation
Substituting \(t = x\) into the vertical motion equation:
\[ y = u_y t - \frac{1}{2} g t^2 \]
\[ y = 2x - \frac{1}{2} \cdot 10 \cdot x^2 \]
\[ y = 2x - 5x^2 \]

### Step 5: Write the final Cartesian equation
Thus, the Cartesian equation of the projectile's path is:
\[ y = 2x - 5x^2 \]

### Conclusion
The correct option corresponding to this equation is option A.

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