Asmall ball with a mass of 60.0 g is loaded into a spring gun,
compressingthe spring by 12.0 cm. when thetrigger is pressed the
ball emerges horizontally from the barrel ata height of 1.40 m
above the floor. itthen strikes the floor after traveling a
horizontal distance of2.20 m. use g = 9.80 m/s2.assuming
all the energy stored in the spring is transferred to theball,
determine the spring constant of the spring.

K=70.58 Nw/m

Explanation:

Conservation Of The Energy

When no losses are considered, like friction or air resistance, the total energy present in a system is constant. Three types of energy are commonly used in simple physics applications:

Kinetic energy:

Potential gravitational energy:

Elastic potential energy

The variables used in the formulas are: m=mass, v=speed, h=height above ground, g=acceleration of gravity, k=spring constant, x=spring compression

When the spring is compressed, the ball is at rest, and only two energies are present, the elastic and the potential. Thus

After the trigger is pressed, all the elastic energy is released and transformed in kinetic and the ball is fired at speed . At this moment, the ball has two energies: Kinetic and potential. The potential energy is the same as before because the ball is still at the same height, so

Equating both energies

Or equivalently

Rearranging

Solving for k

We need to find by using motion conditions. We know the ball reaches a distance d=2.2 m after traveling height of h=1.4 m. We can use these data to find .

The horizontal distance is given by

The vertical height is

Solving for t in [2]

Replacing in [3]

Operating

Solving for

Replacing in [1]

Rearranging

Computing with m=60 gr= 0.06 Kg, x=12 cm = 0.12 m

pythagoras' theorem

square root (-5^2+9^2)

square root (25+81)

square root (106)

angle is tan^-1(9/5)

neg x, pos y