Question

At t= 0, a particle moving in the xy plane with constant acceleration has a velocity of Vi= (3.00i -2.00j) m/s and is at the origin. At t= 3.00 s, the particle's velocity is V = (9.00i + 7.00j) m/s . Find (a) the acceleration of the particle and (b) its coordinates at any time t.

Answer 1

The acceleration of the particle is 2.00i + 3.00j m/s^2. The coordinates of the particle at any time t are given by the expressions v(t) = (3.00i - 2.00j) + (2.00i + 3.00j)t and r(t) = (0i + 0j) + (3.00i - 2.00j)t + (1/2)(2.00i + 3.00j)t^2.

Step-by-step explanation:

To find the acceleration of the particle, we can use the formula:

a = (V - Vi) / t

where a is acceleration, V is final velocity, Vi is initial velocity, and t is time.

Substituting the given values:

We get:

a = ((9.00i + 7.00j) - (3.00i - 2.00j)) / 3.00

Simplifying the expression gives:

a = (6.00i + 9.00j) / 3.00

a = 2.00i + 3.00j m/s^2

So, the acceleration of the particle is 2.00i + 3.00j m/s^2.

To find the coordinates of the particle at any time t, we can use the following equations:

v(t) = Vi + at

r(t) = r(0) + Vit + (1/2)at^2

where v(t) is the velocity at time t, r(t) is the position at time t, r(0) is the initial position, Vi is the initial velocity, a is the acceleration, and t is time.

Substituting the given values:

We get:

v(t) = (3.00i - 2.00j) + (2.00i + 3.00j)t

r(t) = (0i + 0j) + (3.00i - 2.00j)t + (1/2)(2.00i + 3.00j)t^2

So, the coordinates of the particle at any time t are given by the expressions above.

Answer 2

(a) a = (2i + 4.5j) m/s^2

(b) r = ro + vot + (1/2)at^2

Step-by-step explanation:

(a) The acceleration of the particle is given by:

`\vec{a}=\frac{\vec{v}-\vec{v_o}}{t}\\\\`

vo: initial velocity = (3.00i -2.00j) m/s

v: final velocity = (9.00i + 7.00j) m/s

t = 3s

by replacing the values of the vectors and time you obtain:

`\vec{a}=(1)/(3s)[(9.00-3.00)\hat{i}+(7.00-(-2.00))\hat{j}]\\\\\vec{a}=(2\hat{i}+4.5\hat{j})m/s^2`

(b) The position vector is given by:

`\vec{r}=\vec{r_o}+\vec{v_o}t+(1)/(2)\vec{a}t^2`

where vo = (3.00i -2.00j) m/s and a = (2.00i + 4.50j)m/s^2