Question

Suppose that a particle's position is described by r(t)=(t+3)i+(t2+2)j. Give an equation (in the form of a formula involving x and y set equal to 0 ) whose solutions consist of the path of the particle. Find the velocity vector for the particle: v(t)

Answer 1

To find an equation for the path of the particle, set the x and y components of the position vector equal to zero. The velocity vector is found by taking the derivative of the position vector.

Step-by-step explanation:

To find an equation whose solutions consist of the path of the particle, we need to set the x and y components of the position vector equal to zero. In this case, we have (t + 3) = 0 for the x-component and (t^2 + 2) = 0 for the y-component. Solving these equations gives us t = -3 and t = ±√2 for the x and y components, respectively.

So, the equation whose solutions consist of the path of the particle is:

x = -3

y = ±√2

The velocity vector v(t) is the derivative of the position vector r(t). Taking the derivative of r(t) = (t + 3)i + (t^2 + 2)j, we get:

v(t) = 1i + 2tj

Answer 2

The velocity for the particle is (1i+2tj).

Step-by-step explanation:

Given that,

The particle position is

`r(t)=(t+3)i+(t^2+2)j`...(I)

In form of x and y

`x(t)=t+3`....(II)

`y(t)=t^2+2`...(III)

From equation (II)

`t=x-3`

Put the value of t in equation (III)

`y=(x-3)^2+2`

`y=x^2+9-6x+2`

`x^2-6x-y+11=0`

We need to calculate the velocity

Velocity is the rate of change of the position of the particle

`v(t)=(dr)/(dt)`

`\vec{v(t)}=1i+2t j`

Hence, The velocity for the particle is (1i+2tj).