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Chapter 7: Problem 23

You work for a toy company, and you're designing a springlaunched model rocket. The launching apparatus has room for a spring that can be compressed \(14 \mathrm{cm},\) and the rocket's mass is \(65 \mathrm{g} .\) If the rocket is to reach an altitude of \(35 \mathrm{m},\) what should you specify for the spring constant?

Short Answer

Expert verified
The spring constant that should be specified for the rocket to reach an altitude of 35m is approximately \(96428.57 \, N/m\).

Step by step solution

01

Identify Useful Information

The rocket mass \(m\) is 65g or 0.065kg, the spring compression \(d\) is 14cm or 0.14m, the height \(h\) that the rocket must reach is 35m, and gravity \(g\) is 202.3kg/m^2. The spring constant \(k\) is what we need to find.
02

Formulate Relevant Formulas

Use Hooke's Law which states that the force exerted by a spring is proportional to its extension or compression, specified by \(F = -kd\). Also, we need to use the Principle of Conservation of Energy, \(E_{initial} = E_{final}\). Where the initial total energy \(E_{initial}\) consists of the potential energy of the spring \(U_{spring} = 0.5kd^2\) and the final total energy \(E_{final}\) is the potential energy of the rocket at height \(h\), or \(E_{final} = mgh\).
03

Solve for the Unknown

Setting \(E_{initial} = E_{final}\), we have: \(0.5kd^2 = mgh\). Solving for \(k\), we get: \(k = \frac{2mgh}{d^2}\). Substituting all the given values in kilogram and meter units we have: \(k = \frac{2*(0.065)*(9.8)*(35)}{(0.14)^2}\).
04

Calculate

Calculating the above expression for \(k\), we get approximately 96428.57 N/m. So the spring constant \(k\) that should be specified for the rocket to achieve an altitude of 35m must be approximately 96428.57 N/m.

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Most popular questions from this chapter

If the potential energy is zero at a given point, must the force also be zero at that point? Give an example.

The force on a particle is given by \(\vec{F}=A \hat{i} / x^{2},\) where \(A\) is a positive constant. (a) Find the potential-energy difference between two points \(x_{1}\) and \(x_{2},\) where \(x_{1}>x_{2} .\) (b) Show that the potential energy difference remains finite even when \(x_{1} \rightarrow \infty\)

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Blocks with different masses are pushed against a spring one at a time, compressing it different amounts. Each is then launched onto an essentially frictionless horizontal surface that then curves upward, still frictionless (like Fig. 7.21 but without the frictional part). The table below shows the masses, spring compressions, and maximum vertical height each block achieves. Determine a quantity that, when you plot \(h\) against it, should yield a straight line. Plot the data, determine a best-fit line, and use its slope to determine the spring constant. $$\begin{array}{|c|r|r|r|r|r|} \hline \text { Mass } m(\mathrm{g}) & 50.0 & 85.2 & 126 & 50.0 & 85.2 \\ \hline \text { Compression } x(\mathrm{cm}) & 2.40 & 3.17 & 5.40 & 4.29 & 1.83 \\\ \hline \text { Height } h(\mathrm{cm}) & 10.3 & 11.2 & 19.8 & 35.2 & 3.81 \\ \hline \end{array}$$

A 54 -kg ice skater pushes off the wall of the rink, giving herself an initial speed of \(3.2 \mathrm{m} / \mathrm{s}\). She then coasts with no further effort. If the frictional coefficient between skates and ice is \(0.023,\) how far does she go?
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