A bead is moving with a speed of 20 m/s at position A on the track shown in the figure. This track is friction-free, and there is no appreciable air resistance. What is the speed of the bead at point C?
A) 0 m/s
B) 34 m/s
C) 69 m/s
D) 20 m/s
E) We cannot solve this problem without knowing the mass of the bead.

Question

A bead is moving with a speed of 20 m/s at position A on the track shown in the figure. This track is friction-free, and there is no appreciable air resistance. What is the speed of the bead at point C? 
A) 0 m/s 
B) 34 m/s 
C) 69 m/s 
D) 20 m/s 
E) We cannot solve this problem without knowing the mass of the bead.

Quizplus · Accepted Answer

Since the track is friction-free and there is no air resistance, total mechanical energy of the bead is conserved throughout the motion. 
At point A, the mechanical energy of the bead is equal to its kinetic energy: 
$E_{	ext{A}} = \frac{1}{2}mv_{	ext{A}}^2$

At point C, the mechanical energy of the bead is equal to the sum of its kinetic and potential energy:
$E_{	ext{C}} = \frac{1}{2}mv_{	ext{C}}^2 + mgh$

Since mechanical energy is conserved, we can equate these two expressions: 
$E_{	ext{A}} = E_{	ext{C}}$
$\frac{1}{2}mv_{	ext{A}}^2 = \frac{1}{2}mv_{	ext{C}}^2 + mgh$

Since the track is friction-free, the height of point C above point A is equal to the height of point B above point A. Therefore, we can write $h = 2r = 20$ m.

Substituting this value into the energy conservation equation and solving for $v_{	ext{C}}$, we get: 
$v_{	ext{C}} = \sqrt{v_{	ext{A}}^2+2gh} = \sqrt{(20	ext{ m/s})^2+2(9.8	ext{ m/s}^2)(20	ext{ m})} \approx 34	ext{ m/s}$

Therefore, the answer is B: 34 m/s.

A Bead Is Moving with a Speed of 20 M/s