Q: In the experiment, alpha particles follow a hyperbolic trajectory when deflected by a gold nucleus. If the minimum distance between the centers of the alpha particle and the gold nucleus is $$d = 12 \times 10$$^{-14}$$ m, what is the equation of the trajectory of the alpha particle?

$$x = \sqrt{y^2$ + 144 \times 10$$^{-28}$$}$$

Q: Suppose the minimum distance $$d$$ between the alpha particle and the gold nucleus is doubled. How does the constant in the equation $$x = \sqrt{y^2$ + 144 \times 10$$^{-28}$$}$$ change?

It becomes $$576 \times 10$$^{-28}$$

Q: If the equation of the hyperbola is $$x = \sqrt{y^2$ + 144 \times 10$$^{-28}$$}$$, what is the value of $$y$$ when $$x = 13 \times 10$$^{-14}$$?

$$y = \sqrt{(13 \times 10$$^{-14}$$)^2 - 144$$ \times $$10^{-28}$$}$

Q: If the equation of the hyperbola is $$x = \sqrt{y^2$ + 144 \times 10$$^{-28}$$}$$, what is the value of $$x$$ when $$y = 5 \times 10$$^{-14}$$?

$$x = \sqrt{(5 \times 10$$^{-14}$$)^2 + 144$$ \times $$10^{-28}$$}$

Question 1

In the experiment, alpha particles follow a hyperbolic trajectory when deflected by a gold nucleus. If the minimum distance between the centers of the alpha particle and the gold nucleus is $$d = 12 	imes 10$$^{-14}$$ m, what is the equation of the trajectory of the alpha particle?

Accepted Answer

$$x = \sqrt{y^2$ + 144 	imes 10$$^{-28}$$}$$

Question 2

Given the hyperbolic trajectory of an alpha particle deflected by a gold nucleus, what is the minimum distance between their centers, expressed in scientific notation and rounded to the nearest tenth?

Accepted Answer

$$1.2 	imes 10$$^{-14}$$ m

Question 3

Which geometric property of the hyperbola is used to describe the deflection of the alpha particle in the experiment?

Accepted Answer

The focus, where the gold nucleus is located

Question 4

If the equation of the hyperbolic trajectory is $$x = \sqrt{y^2$ + 144 	imes 10$$^{-28}$$}$$, what is the value of $$x$$ when $$y = 0$$?

Accepted Answer

$$x = 12 	imes 10$$^{-14}$$

Question 5

In the context of the experiment, what does the angle $$	heta = 90^\circ$$ represent?

Accepted Answer

The angle at which the alpha particle is deflected by the gold nucleus

Question 6

If the trajectory of the alpha particle passes through a vertex of the hyperbola, what is the significance of this point?

Accepted Answer

It is the point of closest approach to the gold nucleus

Question 7

Suppose the minimum distance $$d$$ between the alpha particle and the gold nucleus is doubled. How does the constant in the equation $$x = \sqrt{y^2$ + 144 	imes 10$$^{-28}$$}$$ change?

Accepted Answer

It becomes $$576 	imes 10$$^{-28}$$

Question 8

Which of the following best describes the path of the alpha particle as it approaches and is deflected by the gold nucleus?

Accepted Answer

It follows a hyperbolic trajectory with the nucleus at a focus

Question 9

If the equation of the hyperbola is $$x = \sqrt{y^2$ + 144 	imes 10$$^{-28}$$}$$, what is the value of $$y$$ when $$x = 13 	imes 10$$^{-14}$$?

Accepted Answer

$$y = \sqrt{(13 	imes 10$$^{-14}$$)^2 - 144$$ 	imes $$10^{-28}$$}$

Question 10

In the experiment, why is the gold nucleus placed at a focus of the hyperbola rather than at the center?

Accepted Answer

Because the focus is the point where the force of repulsion is strongest

Question 11

If the equation of the hyperbola is $$x = \sqrt{y^2$ + 144 	imes 10$$^{-28}$$}$$, what is the distance between the two vertices?

Accepted Answer

$$24 	imes 10$$^{-14}$$

Question 12

Which of the following is a plausible reason for using scientific notation in reporting the minimum distance between the alpha particle and the gold nucleus?

Accepted Answer

The distance is extremely small and scientific notation makes it easier to read and compare

Question 13

If the minimum distance $$d is$$ $$12 	imes 10$$^{-14}$$ m, what is d^2$ in scientific notation?

Accepted Answer

$$144 	imes 10$$^{-28}$$

Question 14

If the equation of the hyperbola is $$x = \sqrt{y^2$ + 144 	imes 10$$^{-28}$$}$$, what is the value of $$x$$ when $$y = 5 	imes 10$$^{-14}$$?

Accepted Answer

$$x = \sqrt{(5 	imes 10$$^{-14}$$)^2 + 144$$ 	imes $$10^{-28}$$}$

Question 15

Which of the following statements is true about the hyperbolic trajectory of the alpha particle?

Accepted Answer

The trajectory is symmetric about the axis passing through the nucleus

Question 16

If the minimum distance $$d is$$ $$12 	imes 10$$^{-14}$$ m, and the equation of the hyperbola is x$$ = \sqrt{$$y^2$$ + $$d^2$$}$$, what is the value of x$ when y = d$?

Accepted Answer

$$x = \sqrt{(12 	imes 10$$^{-14}$$)^2 + (12$$ 	imes $$10^{-14})^2$}$$

Analytic Geometry: Hyperbolas and Particle Trajectories

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