A car in an amusement park roller coaster ride rolls without friction at the top

Question

A car in an amusement park roller coaster ride rolls without friction at the top of a hill. The car begins at a height h from the top of a hill. A the bottom, the car then goes through a vertical loop where the car is upside down at the loop's top. If the radius of the loop is 20.0m, what is the minimum height h such that the car moves around the loop without falling off the track at the top of the loop while upside down? If the height is instead 70m, what is the car's speed at the top of the loop? From a 70m height, what is the magnitude of the car's radial acceleration? From 70m, what is the magnitude of the car's tangential acceleration?

Explanation / Answer

Now we know the velocity (or KE) we need at B so we can use conservation of energy (remember F

N does no work so (Wother=0) to get it

EA=EB

1/2mvA2 +mgyA=1/2mvB2 +mgyB

0+mgh=1/2mgR +mg2R

h=5/2*R

h=50 m

We know that the radial acceleration is v2/R so we need v at C. We can use conservation of energy (remember F does no work so Wother=0) to get it let us follow the step

EA=EC

1/2mvA2 +mgyA=1/2mvc2 +mgyc

0+mgh=1/2mv2 +mgR

v2 =3gR

radical accleration ar=v2/R=3g

ar=29.4 m/s2

Speed calculation

We know that mg+f=mv2/R

f=0 so cancelling m in both sides we get,

g=v2/R

v2=gR=196

v=14 m/s

tangential accelaration is v/t

at=sqrt(gR/t)

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