A vertical spring with a force constant of 250 N/m is compressed 7.5 cm and a 0.30 kg ball placed on top.
The spring is released and the ball flies vertically upward.
How high does the ball go?
h=________cm
(thanks for helping :D)
To calculate how high the ball goes, you can use the principle of conservation of mechanical energy. The potential energy stored in the compressed spring is converted into kinetic energy as the ball is released. As the ball reaches its highest point, all of its initial kinetic energy will have been converted into gravitational potential energy.
To solve for the height the ball reaches, you need to find the difference in potential energy between its initial and final states. The potential energy stored in the compressed spring can be calculated using the formula:
PE_spring = 0.5 * k * x^2
where k is the force constant (250 N/m) and x is the compression distance (7.5 cm or 0.075 m).
PE_spring = 0.5 * 250 N/m * (0.075 m)^2
Now, you need to find the gravitational potential energy at the highest point. The gravitational potential energy formula is:
PE_gravity = m * g * h
where m is the mass of the ball (0.30 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height we want to find.
At the highest point, the ball will momentarily stop, so its kinetic energy will be zero. Therefore, its initial kinetic energy will be equal to the final gravitational potential energy.
KE_initial = PE_gravity
Since kinetic energy is given by:
KE = 0.5 * m * v^2
where v is the velocity, at the highest point, the velocity will be zero.
We can set the initial kinetic energy equal to the gravitational potential energy:
0.5 * m * v^2 = m * g * h
Simplifying:
0.5 * v^2 = g * h
Rearranging the equation to solve for h:
h = (v^2) / (2 * g)
To find v (the velocity), we can use the equation of motion:
v^2 = u^2 - 2 * g * s
where u is the initial velocity, in this case, it is zero since we are only considering the release of the ball from the compressed spring, and s is the distance or compression of the spring (0.075 m).
v^2 = 0 - 2 * 9.8 m/s^2 * 0.075 m
v^2 = -1.47 m^2/s^2
v = -1.21 m/s (taking the positive value since velocity is a scalar quantity)
Now we can substitute the value of v into the previous equation to find h:
h = (1.21 m/s)^2 / (2 * 9.8 m/s^2)
h = 0.0727 m
Converting the answer to centimeters:
h = 0.0727 m * 100 cm/m
h ≈ 7.3 cm
Therefore, the ball reaches approximately 7.3 cm in height.