A child\'s toy consists of a block that attaches to a table, a spring connected

Question

A child's toy consists of a block that attaches to a table, a spring connected to that block, a ball, and a launching ramp. The spring has a spring constant k=500N/m, the ball has a mass m=2kg and the ramp raises a height y=10cm above the table, the surface of which is a height H=80cm above the floor.

Initially, the spring rests at its equilibrium length. The spring then is compressed a distance s=15cm, where the ball is held at rest. The ball is then released, launching it up the ramp. When the ball leaves the launching ramp its velocity vector makes an angle theta=30degrees with respect to the horizontal .

Throughtout this problem, ignore friction and air resistance. PLEASE SHOW WORK

E) Calculate Us the potential energy of the spring at maximum compression.

F) Calculate Vs the speed of the ball at the instant it loses contact with the spring.

G) Calculate the maximum height Hmax with respect to the table reached by the ball.

H) Calculate all energies (kinetic, elastic and/or gravitational), positions and, velocities of the ball that you consider of interest at different positions on the trajectory of the ball. Need 5 quantities calculated.

Explanation / Answer

spring constant k = 500N/m,

the ball has a mass m = 2kg

and the ramp raises a height y = 10cm = 0.1 m above the table,

the surface of which is a height H = 80cm = 0.8 m above the floor.

spring then is compressed a distance s = 15cm = 0.15 m

angle theta=30degrees with respect to the horizontal

E)

Gained potential energy in the form of at maximum compression in spring E = ks2 / 2

E = 1 / 2 X 500 X 0.152

E = 5.625 J

F )

the speed of the ball at the instant it loses contact with the spring

1 / 2 k s2 + m g h = 1 / 2 m v2

1 / 2 X 500 X 0.152 + 2 X 9.8 X 0.8 = 1 / 2 X 2 X v2

5.625 + 15.68 = v2

v2 = 21.305

v = 4.61 m/s

G)

H = v2 / 2g

H = 4.612 / 19.6

H = 21.305 / 19.6

H = 1.08 m

H)

elastic energy = 1/2 ks2 = 0.5 X 500 X 0.15 X 0.15 = 5.625 J

kinetic energy = KE = 1/2mv2 = 0.5 X 2 X 4.61 X 4.61

= 21.25 J

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