Question

4 (a) what is the maximum velocity of the roller coaster car at the lowest point of the track? round the answer to the nearest whole number. a 22 m/s b 175 m/s c 490 m/s d 30,625 m/s (b) determine the pe, ke, and mechanical energy (me) at exactly the middle point between the top of the hill and the very bottom. a pe=30,625 j, ke=0 j, me=30,625 j b pe=0 j, ke=30,625 j, me=61,250 j c pe=61,250 j, 30,625 j, me=30,625 j d pe=30,625 j, ke=30,625 j, me=61,250 j

Explanation:

Step1: Assume conservation of mechanical energy

Mechanical energy $ME = PE+KE$. At the top of the hill (assuming initial velocity $v = 0$), all energy is potential energy $PE=mgh$. At the bottom - most point, all potential energy is converted to kinetic energy $KE=\frac{1}{2}mv^{2}$. But we are not given enough information to calculate from scratch. However, we can make some reasonable estimates. A roller - coaster's speed is usually in the range of a few to tens of meters per second. Option B (175 m/s), C (490 m/s) and D (30625 m/s) are extremely high speeds for a roller - coaster.

Step2: Analyze part (b)

Mechanical energy is conserved in the absence of non - conservative forces (like friction). At the middle point between the top of the hill and the bottom, half of the initial potential energy is converted to kinetic energy. If the total mechanical energy $ME$ is conserved and the initial potential energy at the top is $ME$, at the middle point $PE = KE=\frac{ME}{2}$.

Answer:

(a) A. 22 m/s
(b) D. $PE = 30625\ J, KE = 30625\ J, ME=61250\ J$