Conics

9000106902

Level: 
C
Consider a planet traveling around the Sun on an elliptic trajectory. In the perihelion (the point where the planet is nearest to the Sun) is the distance from the planet to the Sun \(4.5\, \mathrm{AU}\). The excentricity of the ellipse is \(0.5\, \mathrm{AU}\). Find the equation for the trajectory of the planet. Use the coordinate system with center in the Sun and \(x\)-axis along the major axis of the ellipse.
\(\frac{(x-0.5)^{2}} {25} + \frac{y^{2}} {24.75} = 1\)
\(\frac{x^{2}} {25} + \frac{(y-0.5)^{2}} {24.75} = 1\)
\(\frac{x^{2}} {25} + \frac{y^{2}} {24.75} = 1\)
\(\frac{(x-0.5)^{2}} {24.75} + \frac{y^{2}} {25} = 1\)

9000106903

Level: 
C
The motion with a constant acceleration is described by the relation \(s = \frac{1} {2}at^{2}\). Consequently, the graph which shows the distance as a function of time is part of a parabola. Find the directrix of this parabola, if \(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
\(s = -\frac{1} {8}\)
\(s = -1\)
\(s = \frac{1} {8}\)
\(s = 1\)

9000106904

Level: 
C
The motion with a constant deceleration is described by the relation \[ s = v_{0}t -\frac{1} {2}at^{2}. \] Consequently, the graph which shows the distance as a function of time is part of a parabola. Find the focus of this parabola, if \(v_{0} = 16\, \mathrm{m}/\mathrm{s}\) and \(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
\([4;\ 31.875]\)
\([8;\ 31.875]\)
\([4;\ 63.5]\)
\([8;\ 63.5]\)

9000106905

Level: 
C
The motion with a constant deceleration is described by the relation \[ s = v_{0}t -\frac{1} {2}at^{2}. \] Consequently, the graph which shows the distance as a function of time is part of a parabola. Find the vertex equation of this parabola, if \(v_{0} = 8\, \mathrm{m}/\mathrm{s}\) and \(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
\(-\frac{1} {2}(s - 8) = (t - 2)^{2}\)
\(\frac{1} {2}(s + 4) = (t + 2)^{2}\)
\(2(s + 8) = (t + 2)^{2}\)
\(- 2(s + 4) = (t + 2)^{2}\)

9000117701

Level: 
C
A body is thrown at the initial angle \(\alpha = 30^{\circ }\) and the initial velocity \(v_{0} = 20\, \mathrm{m}/\mathrm{s}\). The trajectory of the body is a part of parabola. Find the directrix of this parabola. Hint: The coordinates of the moving body as functions of time are \[ \begin{aligned}x& = v_{0}t\cdot \cos \alpha , & \\y& = v_{0}t\cdot \sin \alpha -\frac{1} {2}gt^{2}. \\ \end{aligned} \] Consider the standard acceleration due to gravity \(g = 10\, \mathrm{m}/\mathrm{s}^{2}\).
\(y = 20\)
\(y = 5\)
\(y = 15\)
\(y = 10\)

9000117702

Level: 
C
The Earth travels around the Sun on an elliptical orbit. The Sun is in the focus of this ellipse. The maximal distance from Earth to the Sun is \(152.1\cdot 10^{6}\, \mathrm{km}\), the minimal distance from Earth to the Sun is \(147.1\cdot 10^{6}\, \mathrm{km}\). Find the length of the semi-minor axis (one half of the length of the shorter axis) and round your answer to the nearest \(10^{4}\, \mathrm{km}\).
\(149.58\cdot 10^{6}\, \mathrm{km}\)
\(2.58\cdot 10^{6}\, \mathrm{km}\)
\(299.21\cdot 10^{6}\, \mathrm{km}\)
\(149.61\cdot 10^{6}\, \mathrm{km}\)

9000117703

Level: 
C
For an isothermal process in an ideal gas the product \(pV \) is constant (Boyle's law). In a pressure-volume diagram which shows \(p\) as a function of \(V \) this law describes a hyperbola (called isotherm). Do we have enough information to identify the asymptotes? If so, find these asymptotes.
\(p = 0\), \(V = 0\)
\(p = V \), \(p = -V \)
\(p = 0\), \(p = V \)
It is not possible to draw any conclusion.

9000117704

Level: 
C
Given physical quantities and laws relating these quantities, identify an answer where the graph which relates these quantities is a part of a hyperbola. (The other quantities are supposed to be constant.)
The pressure (\(p\)) and the area (\(S\)) over which the pressure is distributed, if \(F = p\cdot S\).
The mass (\(m\)) and the kinetic energy (\(E_{k}\)) of a moving body, if \(E_{k} = \frac{1} {2}\cdot m\cdot v^{2}\).
The velocity (\(v\)) and the kinetic energy (\(E_{k}\)) of a moving body, if \(E_{k} = \frac{1} {2}\cdot m\cdot v^{2}\).
The mass (\(m\)) and the potential energy (\(E_{p}\)) in a homogeneous gravitational field, if \(E_{p} = m\cdot g\cdot h\).

9000117705

Level: 
C
Given physical quantities and laws relating these quantities, identify an answer where the graph which relates these quantities is a part of a parabola. (The other quantities are supposed to be constant.)
The electrical work (\(W\)) and the current (\(I\)), if \(W = R\cdot I^{2}\cdot t\).
The mass (\(m\)) and the acceleration (\(a\)) of a moving body, if \(F = m\cdot a\).
The height (\(h\)) and the potential energy (\(E_{p}\)), if \(E_{p} = m\cdot g\cdot h\).
The electrical work (\(W\)) and the time (\(t\)), if \(W = R\cdot I^{2}\cdot t\).

9000117706

Level: 
C
Satellites travel along approximately circular paths. Consider a satellite in the height \(h\) measured from the Earth surface. Further, consider the coordinate system with origin on the Earth surface directly below the satellite and the \(y\)-axis oriented up (to the satellite). The \(x\)-axis is perpendicular to \(y\)-axis and it is in the plane defined by the trajectory of the satellite. Neglect the Earth's rotation and find the equation which describes the path of the satellite. The Earth radius is \(R\).
\(x^{2} + (y + R)^{2} = (R + h)^{2}\)
\(x^{2} + y^{2} = (R + h)^{2}\)
\(x^{2} + (y + R)^{2} = h^{2}\)
\(x^{2} + y^{2} = h^{2}\)