The coordinates of a particle moving in x-y plane at any time t are `(2 t, t^2).` Find (a) the trajectory of the particle, (b) velocity of particle at time t and (c) acceleration of particle at any time t.

Explanation

Let's solve the problem step by step.

### Given:
The coordinates of the particle at any time \( t \) are:
- \( x(t) = 2t \)
- \( y(t) = t^2 \)

### (a) Finding the trajectory of the particle:
To find the trajectory, we need to eliminate the parameter \( t \) from the equations of motion.

1. From the equation \( x = 2t \), we can express \( t \) in terms of \( x \):
\[ t = \frac{x}{2} \]

2. Now, substitute this expression for \( t \) into the equation for \( y \):
\[ y = t^2 = \left(\frac{x}{2}\right)^2 = \frac{x^2}{4} \]

3. Rearranging gives us the equation of the trajectory:
\[ x^2 = 4y \]

### (b) Finding the velocity of the particle at time \( t \):
The velocity \( \mathbf{v} \) of the particle is given by the derivative of the position vector \( \mathbf{r} \) with respect to time \( t \).

1. The position vector \( \mathbf{r} \) can be expressed as:
\[ \mathbf{r} = x \hat{i} + y \hat{j} = 2t \hat{i} + t^2 \hat{j} \]

2. Taking the derivative with respect to \( t \):
\[ \mathbf{v} = \frac{d\mathbf{r}}{dt} = \frac{d(2t)}{dt} \hat{i} + \frac{d(t^2)}{dt} \hat{j} = 2 \hat{i} + 2t \hat{j} \]

### (c) Finding the acceleration of the particle at any time \( t \):
The acceleration \( \mathbf{a} \) of the particle is given by the derivative of the velocity \( \mathbf{v} \) with respect to time \( t \).

1. From the velocity equation, we have:
\[ \mathbf{v} = 2 \hat{i} + 2t \hat{j} \]

2. Taking the derivative with respect to \( t \):
\[ \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d(2)}{dt} \hat{i} + \frac{d(2t)}{dt} \hat{j} = 0 \hat{i} + 2 \hat{j} \]

### Summary of Results:
- (a) The trajectory of the particle is given by \( x^2 = 4y \).
- (b) The velocity of the particle at time \( t \) is \( \mathbf{v} = 2 \hat{i} + 2t \hat{j} \).
- (c) The acceleration of the particle at any time \( t \) is \( \mathbf{a} = 0 \hat{i} + 2 \hat{j} \).