Path traced by a moving particle in space is called trajectory of the particle. Shape of trajectiry is decided by the forces acting on the particle. When a coordinate system is associated with a particle motion, the curve equation in which the particle moves `[y=f(x)]` is called equation of trajectory. It is just giving us the relation among x and y coordinates of the particle i.e. the locus of particle. To find equation of trajectory of a particle, find first x and y coordinates of the particle as a function of time eliminate the time factor.
The position vector of car w.r.t. its starting point is given as `vec(r)=at hat(i)- bt^(2) hat(j)` where a and b are positive constants. The locus of a particle is:-

Options

A. `a^(2)y+bx^(2)=0`
B. `a^(2)y=bx^(2)`
C. `y=b/a^(2)`
D. `ay^(2)=b^(2)x`

Explanation

To find the equation of the trajectory of the particle given its position vector, we will follow these steps:

### Step 1: Identify the position vector components
The position vector of the car with respect to its starting point is given as:
\[ \vec{r} = a t \hat{i} - b t^2 \hat{j} \]
From this, we can identify the x and y coordinates:
- \( x = a t \)
- \( y = -b t^2 \)

### Step 2: Express time in terms of x
From the equation for x, we can express time \( t \) in terms of \( x \):
\[ t = \frac{x}{a} \]

### Step 3: Substitute time into the y equation
Now, we will substitute the expression for \( t \) into the equation for \( y \):
\[ y = -b t^2 \]
Substituting \( t = \frac{x}{a} \):
\[ y = -b \left(\frac{x}{a}\right)^2 \]
This simplifies to:
\[ y = -\frac{b}{a^2} x^2 \]

### Step 4: Rearranging the equation
To express this in a standard form, we can rearrange it:
\[ a^2 y = -b x^2 \]
or equivalently:
\[ a^2 y + b x^2 = 0 \]

### Conclusion
Thus, the locus of the particle is given by the equation:
\[ a^2 y + b x^2 = 0 \]