Answer to Question #169140 in Physics for mahikah

Question #169140

A turtle crawls along a straight line, which we will call the x-axis with the positive direction to

the right. The equation for the turtle’s position as a function of time is π‘₯(𝑑) = 50.0 π‘π‘š +

(2.00 π‘π‘š/s) βˆ’ t (0.0625 π‘π‘š/ s2) t2

.(a) Find the turtle’s initial velocity, initial position, and initiaal acceleration

(b) At what time t is the velocity of the turtle zero?

(c) How long after startingd doesit take the turtle to return to its starting point?

(d) At what times t is the turtle a distance of

10.0 cm from its starting point?

(e) What is the velocity (magnitude and direction) of the turtlea at each of these times?


1
Expert's answer
2021-03-08T08:27:40-0500

(a) Find the first derivative of x(t):


xβ€²(t)=v(t)=2 cm/sβˆ’0.0625 cm/s2β‹…tx'(t)=v(t)=2\text{ cm/s}-0.0625\text{ cm/s}^2 Β·t

The initial velocity at t=0 is 2 cm/s, initial acceleration is 0.0625 cm/s2, initial position at t=0 is 50 cm.

(b) Find when the velocity is 0:


v(t)=0=2βˆ’0.0625t, t=20.0625=32 s.v(t)=0=2-0.0625t,\\\space\\ t=\frac{2}{0.0625}=32\text{ s}.

(c)The starting point is 50 cm:


50=50+2tβˆ’0.0625t2,t=32 s.50=50+2t-0.0625t^2,\\ t=32\text{ s}.

(d) Substitute this into x(t):


10=50+2tβˆ’0.0625t2,0.0625t2βˆ’2tβˆ’40=0,t=45.9 s.10=50+2t-0.0625t^2,\\ 0.0625t^2-2t-40=0,\\ t=45.9\text{ s}.

(We take only positive root because we can't turn back time)

(e) At t=0 it is 2 cm/s, at t=32 is is 0, at t=45.9 it is


v(45.9)=2βˆ’0.0625β‹…45.9=βˆ’0.0869 m/s.v(45.9)=2-0.0625Β·45.9=-0.0869\text{ m/s}.


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