Please help me!

Question

A vertical spring (ignore its mass), whose spring constant is 880 N/m, is attached to a table and is compressed down by 0.150 m. 

a) What upward speed can it give to a 0.340 kg ball when released?

b) How high above its original position (spring compressed) will the ball fly?

Answer 1

Let

k = 880 N/m be the spring constant,

x₀ = 0.150 m be the amount of initial spring compression,

m = 0.340 kg be the mass of the ball,

v (to be calculated) be the maximum speed of the ball,

h (to be calculated) be the maximum height of the ball,

g = 9.81 m/s² be gravitational acceleration.

(b)

Let me first do question b) because it is simpler.

When the ball reaches the maximum height h it has zero kinetic energy,

and its potential energy, mgh, has been obtained from the original spring energy, kx₀²/2,

so the two are equal:

mgh = kx₀²/2

h = kx₀²/2mg

Substituting actual numbers

h = 880×0.15/(2×0.34×9.81) = 2.97 m

(a)

Let h₁ be the height of the ball (above the initial position) when the ball achieves the maximal velocity v.

At the height h₁, the initial spring energy, kx₀²/2,

is partially converted to kinetic energy of the ball, mv²/2,

and partially to potential energy of the ball, mgh₁.

Thus the equation for conservation of energy is

mv²/2 + mgh₁ = kx₀²/2

Solve for v

v = √(kx₀²/m - 2gh₁) (1)

Before we can substitute actual numbers we need to calculate the height h₁.

Let x₁ be the amount of compression at the height h₁, that is,

h₁ = x₀ - x₁

At the outset the spring is compressed by the amount x₀.

When released, it starts accelerating the ball until it reaches the compression x₁,

when it stops accelerating; so that is where the speed is maximum.

At the compression x₁ the upward force of the spring kx₁ equals the downward force of gravity,

for the following reason.

With larger compression the upward force of the spring exceeds the force of gravity,

and that is what is causing the ball's acceleration.

When compression becomes smaller than x₁ then gravity wins over the force of the spring,

causing the deceleration.

To calculate x₁:

kx₁ = mg

x₁ = mg/k

Substituting actual numbers

x₁ = 0.34×9.81/880 = 0.0038 m

Hence

h₁ = x₀ - x₁ = 0.15 - 0.0038 = 0.146

Substituting actual numbers into (1)

v = √(880×0.15²/0.34 - 2×9.81×0.146) = 7.441 m/s