Question

1. a roller coaster starts from rest at the top of an 18 - m hill as shown. the car travels to the bottom of the hill and continues up the next hill that is 10.0 m high. how fast is the car moving at the top of the 10.0 - m hill, if friction is ignored?

Explanation:

Step1: Determine initial and final mechanical energy

Initial state: $E_{i}=mgh_{i}$ (potential - energy only as $v_{i} = 0$), where $h_{i}=18\ m$. Final state: $E_{f}=mgh_{f}+\frac{1}{2}mv_{f}^{2}$, where $h_{f}=10\ m$.

Step2: Apply conservation of mechanical energy

Since there is no friction, $E_{i}=E_{f}$. So, $mgh_{i}=mgh_{f}+\frac{1}{2}mv_{f}^{2}$.

Step3: Simplify the equation

Cancel out the mass $m$ on both sides of the equation: $gh_{i}=gh_{f}+\frac{1}{2}v_{f}^{2}$. Rearrange to solve for $v_{f}$: $\frac{1}{2}v_{f}^{2}=g(h_{i}-h_{f})$.

Step4: Substitute values and calculate

Given $g = 9.8\ m/s^{2}$, $h_{i}=18\ m$, $h_{f}=10\ m$. Then $\frac{1}{2}v_{f}^{2}=9.8\times(18 - 10)$. $\frac{1}{2}v_{f}^{2}=9.8\times8=78.4$. $v_{f}^{2}=2\times78.4 = 156.8$. $v_{f}=\sqrt{156.8}\approx12.52\ m/s$.

Answer:

$12.52\ m/s$