Conics

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 \(R\).
\(x^{2} + (y + R)^{2} = (R + h)^{2}\)
\(x^{2} + y^{2} = (R + h)^{2}\)
\(x^{2} + (y + R)^{2} = h^{2}\)
\(x^{2} + y^{2} = h^{2}\)

9000120005

Level: 
B
The executives of a camp organize a holiday game. For this game it is important that the direct distance kitchen - tent - fireplace is equal for all tents in the camp. Is this information enough to determine the curve passing through all the tents in the camp? Is this curve a conic? If yes, determine which conic.
Yes, all the tents are on an ellipse.
Yes, all the tents are on a circle.
Yes, all the tents are on a parabola.
Yes, all the tents are on a hyperbola.
No, we do not have enough information to draw any conclusion.

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 \(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
\(s = -\frac{1} {8}\)
\(s = -1\)
\(s = \frac{1} {8}\)
\(s = 1\)

9000106901

Level: 
C
A body is thrown at the initial angle \(\alpha = 45^{\circ }\) and the initial velocity \(v_{0} = 10\, \mathrm{m}/\mathrm{s}\). The trajectory of the body is a parabola. Find the equation 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 \(g = 10\, \mathrm{m}/\mathrm{s}^{2}\).
\((x - 5)^{2} = -10\cdot (y - 2.5)\)
\((x - 5)^{2} = 10\cdot (y + 2.5)\)
\(x^{2} = -10\cdot (y - 5)\)
\((x - 5)^{2} = -10\cdot (y + 2.5)\)

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.
\(\frac{(x-0.5)^{2}} {25} + \frac{y^{2}} {24.75} = 1\)
\(\frac{x^{2}} {25} + \frac{(y-0.5)^{2}} {24.75} = 1\)
\(\frac{x^{2}} {25} + \frac{y^{2}} {24.75} = 1\)
\(\frac{(x-0.5)^{2}} {24.75} + \frac{y^{2}} {25} = 1\)

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 \(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
\(-\frac{1} {2}(s - 8) = (t - 2)^{2}\)
\(\frac{1} {2}(s + 4) = (t + 2)^{2}\)
\(2(s + 8) = (t + 2)^{2}\)
\(- 2(s + 4) = (t + 2)^{2}\)