A ball with a mass of #125 g# is projected vertically by a spring loaded contraption. The spring in the contraption has a spring constant of #16 (kg)/s^2# and was compressed by #2/5 m# when the ball was released. How high will the ball go?

Question

A ball with a mass of #125 g# is projected vertically by a spring loaded contraption.  The spring in the contraption has a spring constant of #16  (kg)/s^2# and was compressed by #2/5 m# when the ball was released.  How high will the ball go?

Emily Abbate · Accepted Answer

The height is #=0.52m#

The spring constant is #k=16kgs^-2#

The compression is #x=2/5m#

The potential energy is

#PE=1/2*16*(2/5)^2=1.28J#

This potential energy will be converted to kinetic energy when the spring is released and to potential energy of the ball

#KE_(ball)=1/2m u^2#

Let the height of the ball be #=h #

The acceleration due to gravity is #g=9.8ms^-2#

Then ,

The potential energy of the ball is #PE_(ball)=mgh#

Mass of the ball is #m=0.125kg#

#PE_(ball)=1.28=0.25*9.8*h#

#h=1.28*1/(0.25*9.8)#

#=0.52m#

The height is #=0.52m#

William Audy · Answer

undefined To find the maximum height the ball will reach, we can use the conservation of mechanical energy principle. The potential energy stored in the compressed spring is converted into kinetic energy when the ball is released, and then into gravitational potential energy at the highest point of its trajectory.

The potential energy stored in the compressed spring is given by: PE = (1/2) * k * x^2, where k is the spring constant and x is the compression distance.

The kinetic energy of the ball when it is released is equal to the potential energy stored in the compressed spring: KE = PE.

At the highest point of the ball's trajectory, all of its kinetic energy is converted into gravitational potential energy: KE = m * g * h, where m is the mass of the ball, g is the acceleration due to gravity, and h is the maximum height.

Setting the potential energies equal and solving for h: (1/2) * k * x^2 = m * g * h.

Substituting the given values: (1/2) * 16 * (2/5)^2 = 0.125 * 9.8 * h.

Solving for h: h = (0.5 * 16 * (2/5)^2) / (0.125 * 9.8).

h ≈ 0.816 m.

So, the maximum height the ball will reach is approximately 0.816 meters.