Question

A particle initially located at the origin has an acceleration of a 2.00j m/s2 and an initial velocity of v-6.00i m/s. (a) Find the vector position of the particle at any time t (where t is measured in seconds). ti+ t2 j) m (b) Find the velocity of the particle at any time t. I + tj) m/s (c) Find the coordinates of the particle at t 5.00 s. (d) Find the speed of the particle at t 5.00 s. m/s

Answer 1

• 
  `\vec{r}(t) = (-6.00 (m)/(s) \ t , (1)/(2) \ 2.00  \ (m)/(s^2) \ t^2 )`
• 
  `\vec{v}(t) = (-6.00 (m)/(s), 2.00  \ (m)/(s^2) \ t )`
• 
  `\vec{r}( 5.00 \ s) = (-30.00 \ m , 25.00 \ m )`
• 
  `| \vec{v} (5.00 \ s) | =11.66 (m)/(s)`

Step-by-step explanation:

The initial position of the particle,
`\vec{r}_0`, is:

`\vec{r}_0 = (0,0)`

the initial velocity is:

`\vec{v}_0 = - 6.00 \ (m)/(s) \hat{i} = (-6.00 (m)/(s),0)`

and the initial acceleration:

`\vec{a} = 2.00  \ (m)/(s^2) \ \hat{j} = ( 0, 2.00  \ (m)/(s^2))`

a

The position
`\vec{r}` at time t is

`\vec{r}(t) = \vec{r}_0 + \vec{v}_0  \ t   + (1)/(2) \ \vec{a} \ t^2`

So, for our problem is:

`\vec{r}(t) = (0,0) + (-6.00 (m)/(s),0)  \ t   + (1)/(2) \ ( 0, 2.00  \ (m)/(s^2)) \ t^2`

`\vec{r}(t) = (0 -6.00 (m)/(s) \ t , 0 +  (1)/(2) \ 2.00  \ (m)/(s^2) \ t^2 )`

`\vec{r}(t) = (-6.00 (m)/(s) \ t , (1)/(2) \ 2.00  \ (m)/(s^2) \ t^2 )`

b

The velocity
`\vec{v}` at time t is

`\vec{v}(t) = \vec{v}_0  + \vec{a} \ t`

So, for our problem is:

`\vec{v}(t) = (-6.00 (m)/(s),0)   + ( 0, 2.00  \ (m)/(s^2)) \ t`

`\vec{v}(t) = (-6.00 (m)/(s), 2.00  \ (m)/(s^2) \ t )`

c

At time 5.00 seconds the position will be:

`\vec{r}( 5.00 \ s) = (-6.00 (m)/(s) \ 5.00 \ s , (1)/(2) \ 2.00  \ (m)/(s^2) \ (5.00 \ s ) ^2 )`

`\vec{r}( 5.00 \ s) = (-30.00 \ m , 25.00 \ m )`

d

and the speed will be :

`| \vec{v} (5.00 \ s) | = |(-6.00 (m)/(s), 2.00  \ (m)/(s^2) \ 5.00 \ s) |`

`| \vec{v} (5.00 \ s) | = |(-6.00 (m)/(s), 10.00  \ (m)/(s)) |`

`| \vec{v} (5.00 \ s) | = \sqrt{ (-6.00 (m)/(s))^2 + (10.00  \ (m)/(s)))^2 }`

`| \vec{v} (5.00 \ s) | = \sqrt{ 36.00 (m^2)/(s^2) + 100.00  \ (m^2)/(s^2)}`

`| \vec{v} (5.00 \ s) | =11.66 (m)/(s)`