Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let the upstream breadcrumb's position act as the origin, so the position at time [tex]t[/tex] of upstream crumb [tex](B_U)[/tex], duck [tex](D)[/tex], and downstream crumb [tex](B_D)[/tex] are, respectively,
[tex]x_{B_U} = \left(2.0\dfrac{\rm m}{\rm s}\right) t[/tex]
[tex]x_D = d\,\mathrm m + v_D t[/tex]
[tex]x_{B_D} = 2d\,\mathrm m + \left(2.0\dfrac{\rm m}{\rm s}\right) t[/tex]
where [tex]d[/tex] is the distance (in m) between the duck and either crumb at the start, and [tex]v_D[/tex] is the velocity of the duck relative to the Earth.
To get both breadcrumbs, the duck has to choose between chasing the upstream or downstream crumb first.
• If the upstream crumb is chosen first, then [tex]v_D[/tex] is 2.0 - 1.0 = 1.0 m/s. Find the time it takes for the duck to reach the upstream crumb:
[tex]x_{B_U} = x_D \implies \left(2.0\dfrac{\rm m}{\rm s}\right) t = d + \left(1.0\dfrac{\rm m}{\rm s}\right) t \\\\ \implies t = d \, \mathrm s[/tex]
In this time, the duck covers a distance of
[tex]x_D = d\,\mathrm m + \left(1.0\dfrac{\rm m}{\rm s}\right) (d\, \mathrm s) = 2d \, \mathrm m[/tex]
while the downstream crumb will have traveled a distance of
[tex]x_{B_D} = 2d\,\mathrm m + \left(2.0\dfrac{\rm m}{\rm s}\right) (d \,\mathrm s) = 4d \,\mathrm m[/tex]
putting a total distance of [tex]2d+4d=6d\,\mathrm m[/tex] between the duck and the remaining crumb.
Take the duck's new position to be the new origin. When the duck turns around and travels with the current, its speed [tex]v_D[/tex] will be 2.0 + 1.0 = 3.0 m/s. Then at time [tex]t[/tex], the duck and remaining crumb have position relative to the new origin given by
[tex]x_D' = \left(3.0\dfrac{\rm m}{\rm s}\right) t[/tex]
[tex]x_{B_D}' = 6d \,\mathrm m + \left(2.0\dfrac{\rm m}{\rm s}\right) t[/tex]
Find the time the duck needs to catch up to the downstream crumb:
[tex]x_D' = x_{B_D}' \implies \left(3.0\dfrac{\rm m}{\rm s}\right) t = 6d\,\mathrm m + \left(2.0\dfrac{\rm m}{\rm s}\right) t \\\\ \implies t = 6d \,\mathrm s[/tex]
Then the total time the duck needs to get both crumbs is [tex]d + 6d = 7d \,\mathrm s[/tex].
• If instead the downstream crumb is chased down first, we start with [tex]v_D[/tex] = 3.0 m/s, so that
[tex]x_D = x_{B_D} \implies d\,\mathrm m + \left(3.0\dfrac{\rm m}{\rm s}\right) t = 2d\,\mathrm m + \left(2.0\dfrac{\rm m}{\rm s}\right) t \\\\ \implies t = d \, \mathrm s[/tex]
In this time, the duck covers a distance of
[tex]x_D = d\,\mathrm m + \left(3.0\dfrac{\rm m}{\rm s}\right) (d\,\mathrm s) = 4d \,\mathrm m[/tex]
while the upstream crumb moves
[tex]x_{B_U} = \left(2.0\dfrac{\rm m}{\rm s}\right) (d\,\mathrm s) = 2d \,\mathrm m[/tex]
The distance between the duck and remaining crumb is then [tex]4d-2d=2d\,\mathrm m[/tex]. Now the duck turns around with speed [tex]v_D[/tex] = 1.0 m/s. If the crumb's new position is taken to be the origin, then
[tex]x_{B_U}' = \left(2.0\dfrac{\rm m}{\rm s}\right) t[/tex]
[tex]x_D' = 2d\,\mathrm m + \left(1.0\dfrac{\rm m}{\rm s}\right) t[/tex]
Find the time it takes for the duck to get the last crumb:
[tex]x_{B_U}' = x_D' \implies \left(2.0\dfrac{\rm m}{\rm s}\right)t = 2d\,\mathrm m + \left(1.0\dfrac{\rm m}{\rm s}\right) t \\\\ \implies t = 2d\,\mathrm s[/tex]
Then the duck can get both crumbs in a total of [tex]d+2d=3d\,\mathrm s[/tex].
To get both crumbs in the shortest time, the duck should go after the downstream crumb first. This path over 2 times faster than the other.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.