Question #297341

{F} The equation for a displacement 𝑠(𝑚), at a time 𝑡(𝑠) by an object starting at a displacement of 𝑠0 (𝑚), with an initial velocity 𝑢(𝑚𝑠 −1 ) and uniform acceleration 𝑎(𝑚𝑠 −2 ) is: 𝑠 = 𝑠0 + 𝑢𝑡 + 1 2 𝑎𝑡 2 A projectile is launched from a cliff with 𝑠0 = 30 𝑚, 𝑢 = 55 𝑚𝑠 −1 and 𝑎 = −10 𝑚𝑠 −2 . The tasks are to: a) Plot a graph of distance (𝑠) vs time (𝑡) for the first 10s of motion. b) Determine the gradient of the graph at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. c) Differentiate the equation to find the functions for: i) Velocity (𝑣 = 𝑑𝑠 𝑑𝑡) ii) Acceleration (𝑎 = 𝑑𝑣 𝑑𝑡 = 𝑑 2 𝑠 𝑑𝑡2 ) d) Use your results from part c to calculate the velocity at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. e) Compare your results for part b) and part d). f) Find the turning point of the equation for the displacement 𝑠 and using the second derivative verify whether it is a maximum, minimum or point of inflection. g) Compare your results from f) with the graph you produced in a).


1
Expert's answer
2022-02-14T16:53:40-0500
s(t)=30+55t10t22,ms(t)=30+55t-\dfrac{10t^2}{2}, m

a)


s(t)=30+55t5t2,0t10s(t)=30+55t-5t^2, 0\le t\le10


b)


grad s=s2s1t2t1grad \ s=\dfrac{s_2-s_1}{t_2-t_1}

t=2,s(2)=120t=2 , s(2)=120


t=0,s=50t=0 , s=50


grad st=2=1205020=35(m/s)grad \ s|_{t=2}=\dfrac{120-50}{2-0}=35(m/s)

t=6,s(6)=180t=6 , s(6)=180


t=0,s=210t=0 , s=210

grad st=6=18021060=5(m/s)grad \ s|_{t=6}=\dfrac{180-210}{6-0}=-5(m/s)


c)

i)


v(t)=dsdt=5510t,m/sv(t)=\dfrac{ds}{dt}=55-10t, m/s

ii)


a(t)=dvdt=d2sdt2=10 m/s2a(t)=\dfrac{dv}{dt}=\dfrac{d^2s}{dt^2}=-10\ m/s^2

d)


v(2)=5510(2)=35(m/s)v(2)=55-10(2)=35(m/s)

v(6)=5510(6)=5(m/s)v(6)=55-10(6)=-5(m/s)

e) The results are the same.


f) The turning point of the equation for the displacement

dsdt=0=>5510t=0\dfrac{ds}{dt}=0=>55-10t=0

t=5.5st=5.5 s

s(5.5)=30+55(5.5)5(5.5)2=181.25(m)s(5.5)=30+55(5.5)-5(5.5)^2=181.25(m)

Turning point is (5.5,181.25)(5.5, 181.25)


d2sdt2=10<0\dfrac{d^2s}{dt^2}=-10<0

Point (5.5,181.25)(5.5, 181.25) is a maximum.


g) The results are the same.



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