A ball with a mass of #70 g# is projected vertically by a spring loaded contraption. The spring in the contraption has a spring constant of #49 (kg)/s^2# and was compressed by #7/6 m# when the ball was released. How high will the ball go?

Question

A ball with a mass of #70 g# is projected vertically by a spring loaded contraption.  The spring in the contraption has a spring constant of #49  (kg)/s^2# and was compressed by #7/6 m# when the ball was released.  How high will the ball go?

Leo Abeyta · Accepted Answer

The height is #=48.61m#

The spring constant is #k=49kgs^-2#

The compression is #x=7/6m#

The potential energy in the spring is

#PE=1/2kx^2=1/2*49*(7/6)^2=33.35J#

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) m#

Then ,

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

The mass of the ball is #=0.070kg#

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

#PE_(ball)=33.35=0.070*9.8*h#

The height is #h=33.35*1/(0.070*9.8)#

#=48.61m#

William Audy · Answer

undefined To determine the maximum height the ball will reach, we can use the conservation of mechanical energy. At the maximum height, all of the kinetic energy of the ball will be converted into gravitational potential energy.

First, calculate the potential energy stored in the compressed spring:

Potential energy (spring) = 1/2 * k * x^2

Where:
k = spring constant (49 kg/s^2)
x = compression distance (7/6 m)

Potential energy (spring) = 1/2 * 49 * (7/6)^2

Next, use the potential energy of the spring to find the kinetic energy of the ball just before it is released:

Potential energy (spring) = Kinetic energy (ball)

Then, use the kinetic energy to find the velocity of the ball just before it is released:

Kinetic energy (ball) = 1/2 * m * v^2

Where:
m = mass of the ball (70 g)
v = velocity of the ball just before release

After finding the velocity, use it to determine the maximum height the ball will reach using the equation for gravitational potential energy:

Gravitational potential energy = m * g * h

Where:
m = mass of the ball (70 g)
g = acceleration due to gravity (9.8 m/s^2)
h = maximum height

Once you have the maximum height, you can calculate it.