Question

I'm kind of confused on this question and #6 is just the continuation of the first question.

Answer 1

We are given that a marble is launched from a spring and we are asked to determine the initial speed as the marble leaves the spring. To do that we will consider the energy of the marble when it leaves the spring.

The spring has elastic potential energy due to its compression. The amount of energy is given by the following equation:

`E_s=(1)/(2)ky^2`

Where

`\begin{gathered} k=\text{ constant of the spring} \\ y=\text{ compression} \\ E_s=\text{ energy of the spring} \end{gathered}`

Now, all this energy is transferred to the marble and it is converted into kinetic energy of the marble. The kinetic energy is given by:

`K=(1)/(2)mv^2`

Where:

`\begin{gathered} K=\text{ kinetic energy} \\ m=\text{ mass} \\ v=\text{ velocity} \end{gathered}`

Due to conservation of energy, these quantities are equal, therefore:

`E_s=K`

Substituting the equations:

`(1)/(2)ky^2=(1)/(2)mv^2`

We can cancel out the 1/2:

`ky^2=mv^2`

Now we solve for "v", first by dividing both sides by the mass:

`(ky^2)/(m)=v^2`

Now we take the square root to both sides:

`\sqrt{(ky^2)/(m)}=v`

Simplifying we get:

`y\sqrt[]{(k)/(m)}=v`

Now. In the case of Neil, we have:

`y_{\text{neil}}\sqrt[]{(k_(neil))/(m)}=v_{\text{Neil}}`

Now we substitute the corresponding values:

Now we need to convert the 14 cm into meters. To do that we will use the following conversion factor:

Multiplying by the conversion factor we get:

We also need to convert the 4.5 g into kg. To do that we will use the following conversion factor:

Multiplying by the conversion factor we get:

Now we substitute the new quantities:

`(0.14m)\sqrt[]{(50.8(N)/(m))/(0.0045kg)}=v_{\text{Neil}}_{}`

Now we solve the operations:

`14.89(m)/(s)=v_{\text{Neil}}`

Therefore, Neil's marble has an initial speed of 14.89 m/s. The same procedure is used to determine the initial speed of Gus's marble but substituting the corresponding spring constant and compression.