Question 1161488
Let {{{v[1]}}}, {{{v[2]}}}, {{{v[3]}}}, and {{{v[4]}}} be the velocities of roller coaster at different points, and {{{h[1]}}},{{{ h[2]}}}, {{{h[3]}}}, and {{{h[4]}}} be the height of the points respectively.


Write the given values of question.

{{{h[1]=32m}}}
{{{h[2]=0}}}
{{{h[3]=26m}}}
{{{h[4]=14m}}}

we know start is at point {{{h[1]}}}, so {{{highlight(v[1]=0)}}}

Write the expression for conservation of energy between points 1 and 2 and substitute the above values.


{{{K*E[1]+P*E[1]=K*E[2]+P*E[2]}}}

{{{0+mgh[1]=(1/2)mv[2]^2+0}}}...........simplify

{{{gh[1]=(1/2)v[2]^2}}}

{{{v[2]^2=gh[1]/(1/2)}}}

{{{v[2]^2=2gh[1]}}}

{{{v[2]=sqrt(2gh[1])}}}........plug in {{{g=9.81(m/s^2)}}} and {{{h[1]=32m}}}

{{{v[2]=sqrt(2*9.81(m/s^2)*32m)}}}

{{{v[2]=sqrt((627.84m^2)/s^2)}}}

{{{highlight(v[2]=25.0567(m/s))}}}



{{{K*E[2]+P*E[2]=K*E[3]+P*E[3]}}}

{{{(1/2)mv[2]^2+mgh[2]=(1/2)mv[3]^2+mgh[3]}}}....simplify and substitute known

{{{(1/2)(25.0567(m/s))^2+9.81(m/s^2)*0=(1/2)v[3]^2+9.81(m/s^2)*26m}}}

{{{(313.919m^2)/s^2=(1/2)v[3]^2+(255.06m^2)/s^2}}}

{{{(313.919m^2)/s^2-(255.06m^2)/s^2=(1/2)v[3]^2}}}

{{{(58.859m^2)/s^2=(1/2)v[3]^2}}}

{{{v[3]^2=(58.859m^2/s^2)/(1/2)}}}

{{{v[3]^2=(117.718m^2)/s^2}}}

{{{v[3]=sqrt((117.718m^2)/s^2)}}}

{{{highlight(v[3]=10.8498(m/s))}}}



{{{K*E[3]+P*E[3]=K*E[4]+P*E[4]}}}

{{{(1/2)mv[3]^2+mgh[3]=(1/2)mv[4]^2+mgh[4]}}}....simplify 

{{{(1/2)v[3]^2+gh[3]=(1/2)v[4]^2+gh[4]}}}......substitute known

{{{(58.8591m^2)/s^2+(255.06m^2)/s^2=(1/2)v[4]^2+(137.34m^2)/s^2}}}

{{{(313.919m^2)/s^2- (137.34m^2)/s^2 =(1/2)v[4]^2}}}

{{{(176.579m^2)/s^2=(1/2)v[4]^2}}}

{{{v[4]^2=(176.579m^2/s^2)/(1/2)}}}

{{{v[4]^2=(353.158m^2)/s^2}}}

{{{v[4]=sqrt((353.158m^2)/s^2)}}}

{{{highlight(v[4]=18.7925(m/s))}}}