Question

A particle slides back and forth on a frictionless track whose height as a function of horizontal position x is given by y=ax2, where a = 0.81 m−1 .If the particle's maximum speed is 8.4 m/s , find the turning points of its motion. (x1,x2) in m

Determine the escape speed from (a) Jupiter's moon Callisto, with mass 1.07×1023kg and radius 2.40 Mm, and (b) a neutron star, with the Sun's mass crammed into a sphere of radius 5.60 km .

Answer 1

1) A particle slides back and forth on a frictionless track whose height as a function of horizontal position x is given by y = ax^2, where a = 0.81 m^-1. If the particle's maximum speed is 8.4m/s, find the turning points of its motion.

This is like a carnival ride. The shape of track is a parabola. The equation of the parabola is y = 0.81 * x^2

The x-coordinate of the lowest point of the ride is 0.

y = 0.81 * 0^2 = 0

The point (0, 0) is the lowest point of the parabola shaped carnival ride.

The particle moves down from the highest point on the left side of parabola to the lowest point and then up to the highest point on the right side. The velocity increases as the particle moves down, and the velocity decreases as the particle moves up. The maximum velocity occurs at the lowest point. At the highest point, the velocity is 0 m/s

As particle moves down, the potential energy decreases and the kinetic energy increases. The maximum kinetic energy occurs at the lowest point. The kinetic energy at the highest point = 0 J.

Decrease of PE = mass * g * h

Increase of KE = ½ * mass * v^2

Decrease of PE = Increase of KE

mass * g * h = ½ * mass * v^2

g * h = ½ * v^2

9.8 * h = ½ * 8.4^2 = 35.28

h = 35.28 ÷ 9.8 = 3.6m

This is y-coordinate of the highest point.

y = 0.81 * x^2

3.36 = 0.81 * x^2

x^2 = 3.36 ÷ 0.81

x = (3.36 ÷ 0.81)^0.5 = ±2.11

This is x-coordinate of the highest point.

The 2 turning points = (2.11, 3.6) and (-2.11, 3.6) m

If only x-coordinates is needed, then (-2.11,2.11) m