the position vector of a particle r ⃗=(t^2-1) i ̂+2tj ̂ . determine the trajectory of the particle in XY plane

We are given the position vector of a particle:

r⃗ = (t² - 1) î + 2t ĵ

Step 1: Express in Component Form
The given position vector can be written in terms of x and y coordinates as:

x = t² - 1
y = 2t

Step 2: Express t in Terms of y
From the equation y = 2t, solve for t:

t = y / 2

Step 3: Substitute t into the x Equation
Substituting t = y / 2 into x = t² - 1:

x = (y / 2)² - 1
x = y² / 4 - 1

Step 4: Rearrange the Equation
Rearrange the equation to get the trajectory of the particle:

x + 1 = y² / 4
4(x + 1) = y²
y² = 4(x + 1)

Step 5: Identify the Trajectory
The equation y² = 4(x + 1) represents a parabola opening to the right with vertex at (-1, 0).

Final Answer:
The trajectory of the particle in the XY-plane is a parabola given by the equation:

y² = 4(x + 1)