You collect some more data on that horse at a later time interval, but now you are measuring thehorse’s velocity, not its position, using a radar gun and you found that the data was described by thefunction: ??x( ?? )= (10 ??/??) +( 2 ??/??^3)t^2 Once again, you have chosen a particular spot on the trackto be your origin and started your clock (t = 0) when you started collecting the data (at which point thehorse was at -2 m in your coordinate system). What is the position of the horse as a function of timeduring this interval?

Note: The position of the horse is x = 2m. There is a typing error in the question. Otherwise, The solution to cubic equation holds a negative value of time t.

Given:

- v(t) = 10 + 2*t^2 (radar gun)

- x*(t) = 10 + 5t^2 + 3t^3 (our coordinate)

Find:

-The position x of horse as a function of time t in radar system.

-The position of the horse at x = 2m in our coordinate system

Solution:

- The position of horse according to radar gun:

v(t) = dx / dt = 10 + 2*t^2

- Separate variables:

dx = (10 + 2*t^2).dt

- Integrate over interval x = 0 @ t= 0

x(t) = 10t + (2/3)*t^3

- time @ x = 2 :

2 = 10t + (2/3)*t^3

0 = 10t + (2/3)*t^3 + 2

- solve for t:

t = 0.1875 s

- Evaluate x* at t = 0.1875 s

x*(0.1875) = 10 + 5(0.1875)^2 + 3(0.1875)^3

x*(0.1875) = 10.18 m