On a map of a city, the town hall is represented by a point and a river through the
city by a straight line. There are places in the city with the property that the direct
distance from each place to the town hall is equal to the direct distance to the river.
In the following list identify a curve which can be used to join all these places.
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 \(E_{p} = m\cdot g\cdot h\).
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 \(W = R\cdot I^{2}\cdot t\).
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 \(R\).
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.
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
\(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
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
\(g = 10\, \mathrm{m}/\mathrm{s}^{2}\).
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.
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
\(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).