Answer: 20.765 m/s
Explanation:
This problem can be solved by the conservation of energy principle, this means the initial energy [tex]E_{o}[/tex] must be equal to the final energy [tex]E_{f}[/tex]:
[tex]E_{o}=E_{f}[/tex] (1)
Where each energy is the sum of kinetic energy [tex]K[/tex] and potential energy [tex]U[/tex]:
[tex]K_{o}+U_{o}=K_{f}+U_{f}[/tex] (2)
Where:
[tex]K_{o}=\frac{1}{2}mV_{o}^{2}[/tex]
Being [tex]m[/tex] your mass and [tex]V_{o}=0 m/s[/tex] your initial velocity, since the roller coaster sterted from rest.
[tex]U_{o}=mgh_{o}[/tex]
Being [tex]g=9.8 m/s^{2}[/tex] the acceleration due gravity and [tex]h_{o}=25 m[/tex] your initial height
[tex]K_{f}=\frac{1}{2}mV_{f}^{2}[/tex]
Being [tex]V_{f}[/tex] your final velocity
[tex]U_{f}=mgh_{f}[/tex]
Being [tex]h_{f}=3 m[/tex] your final height
Rewritting (2):
[tex]\frac{1}{2}mV_{o}^{2}+mgh_{o}=\frac{1}{2}mV_{f}^{2}+mgh_{f}[/tex] (3)
[tex]mgh_{o}=m(\frac{1}{2}V_{f}^{2}+gh_{f})[/tex] (4)
Isolating [tex]V_{f}[/tex]:
[tex]V_{f}=\sqrt{2g(h_{o}-h_{f})}[/tex] (5)
[tex]V_{f}=\sqrt{2(9.8 m/s^{2})(25 m-3 m)}[/tex] (6)
Finally:
[tex]V_{f}=20.765 m/s[/tex] This is your spedd when you arrive at 3 m height
