Question

A rocket is fired from rest at x = 0 and travels along a parabolic trajectory described by y 2 = [120(10 3 )x] m. If the x component of acceleration is a x = ( 4 1 t 2 ) , m/s 2 , where t is in seconds, determine the magnitude of the rockets velocity and acceleration when t = 10 s.

Answer 1

The velocity is 1003.5 m/s and the acceleration is 103.1 m/
`s^(2)`.

Step-by-step explanation:

We need to find the parameter equation of x. To find it, we will need to integrate the x-component acceleration equation given to us, twice. Acceleration is:

a =
`(dv)/(dt)`

dv = adt

∫dv = ∫adt

`\int\limits^v_0 \, dv` =
`\int\limits^t_0 {(1)/(4)t^(2) } \, dt`

v =
`(1)/(12) t^(3) m/s`.

Velocity is:

v =
`(dx)/(dt)`

dx=vdt

Again, integrate both sides:

∫dx = ∫vdt

`\int\limits^x_0 dx` =
`\int\limits^t_0 {(1)/(12) t^(3) } \, dt`

x =
`(1)/(48) t^(4) m`

Substitute our x equation into our parameter equation of y.

y
`^(2)` = [120(
`10^(3)`)x] m

y
`^(2)` = [120(
`10^(3)`)(
`(1)/(48)`)(
`t^(4)`) ]

(take the square root of both sides and simplify)

y = 50
`t^(2)`

Now that we can represent our equation with respect to time, we can take the derivative to figure out the velocity. Remember that taking the derivative of a position function gives us the velocity function.

y = 50
`t^(2)`

vy= y = 100t

Let us write down the two equations for velocity we found:

vx =
`(1)/(12) t^(3) m/s`

vy = 100t m/s

At t = 10 s:

vx =
`(1)/(12) 10^(3)` = 83.3 m/s

vy = 100(10)=1000 m/s

The magnitude of velocity is:

v =
`\sqrt{(vx)^(2)+(vy)^(2) }`

v =
`\sqrt{(83.3)^(2)+(1000)^(2) } = 1003.5 m/s`

To figure out the acceleration, we need to figure out ay which can be found by taking the derivative of the vy equation,

vy = 100t m/s

ay = vy = 100 m/
`s^(2)`

Since ax is given to us in the question, we have the following:

ax =
`((1)/(4)t^(2)) m/s^(2)`

ay = 100 m/
`s^(2)`

At t = 10 s:

ax =
`((1)/(4)10^(2))} =25 m/s^{2`

ay = 100 m/
`s^(2)`

The magnitude of acceleration is equal to:

a =
`\sqrt{(ax)^(2)+(ay)^(2) }`

a =
`\sqrt{(25)^(2)+(100)^(2) }`

a= 103.1 m/
`s^(2)`

Final Answers:

v = 1003.5 m/s

a= 103.1 m/
`s^(2)`