Geometric and physical applications

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Problem: A sandwich, as usual, has fallen down with the buttered side on the horizontal heated surface, having the initial speed of translational motion \( V_{0} \). Assuming that the force of resistance to the motion of the sandwich along the surface was proportional to the momentum, i.e. \( F_{\text {res. }}=-k m V \), where \( k \) is the coefficient of proportionality, find the space passed by the sandwich until the butter is completely melted. The sandwich has left a fixed trace of melted butter of constant thickness and width on the surface, and the speed of the slide to the indicated moment of the time has decreased twice. The initial mass of the butter on the sandwich is equal to the mass of a slice of bread.

8.1.4.31 Geometric and physical applications

3.21 $

Problem: A material point of mass \( \mathrm{m} \) is attracted by the center \( O \) with a force, proportional to the distance. The movement starts from point \( A \) at a distance \( a \) from the center with an initial speed \( v_{0} \), perpendicular to the segment \( O A \). Find the trajectory of motion.

8.1.4.1 Geometric and physical applications

3.7 $

Problem: A material point of mass \( m \) moves linearly under the influence of repulsive force from a fixed center, proportional to the distance from the point to the center, (the proportionality coefficient \( k>0 \) ). The drag force of the environment is proportional to the velocity (the proportionality coefficient \( \lambda>0 \) ). At the initial moment the point is at a distance \( a \) from the center, the velocity is equal to \( v_{0} \) is directed along the straight line, connecting the point with the center. Find the law of motion of the point.

8.1.4.2 Geometric and physical applications

1.98 $

Problem: A material point of mass \( m \) moves linearly under the action of a force of attraction to a fixed center, proportional to the distance from the point to the center (the proportionality coefficient \( k>0 \) ). The drag force of the environment is proportional to the velocity (the proportionality coefficient \( \lambda>0) \). At the initial moment, the distance from the point to the center is equal to \( a \), and the velocity is directed along the straight line connecting the point with the center, and equal to \( v_{0} \). Find the law of motion of a point provided that \( \lambda^{2}<4 m k \).

8.1.4.3 Geometric and physical applications

3.21 $

Problem: Write the differential equation of the hyperbola \( x^{2}-y^{2}=2 a x \).

8.1.4.4 Geometric and physical applications

0 $

Problem: Write a differential equation of the family of curves, in which the segment of any tangent, closed between the coordinate axes, is divided by the tangent point \( M(x, y) \) in relation to \( |A M|:|M B|=2: 1 \), where \( A \) is the point of intersection of the tangent with the \( O y \) axis, \( B \) is the point of intersection with the \( O x \) axis. clip2net.com

8.1.4.5 Geometric and physical applications

0 $

Problem: Using the method of isocline make the approximate family of integral curves of the equation \( y^{\prime}=x+y \). clip2net.com

8.1.4.6 Geometric and physical applications

0 $

Problem: Find the equation of the curve, passing through the point \( (1 / 2,-1) \), if the length of the segment of the semi-axis of the abscissa, cut off by its tangent, is equal to the square of the abscissa of the point of tangency.

8.1.4.7 Geometric and physical applications

1.73 $

Problem: Find the equation of the curve, passing through the origin of the coordinates, if for any segment \( [\alpha, x] \) the area of the curvilinear trapezoid, bounded by the corresponding arc of this curve, is equal to the cube of the ordinate of the endpoint of the arc. clip2net.com

8.1.4.8 Geometric and physical applications

1.48 $

Problem: Find orthogonal trajectories of the family of curves \( y=a x^{2} \).

8.1.4.9 Geometric and physical applications

0.99 $

Problem: The rate of heat loss of a body is proportional to the difference in the temperatures between the body and its environment (Newton's law). Find the dependence of temperature \( T \) at time \( t \), if the body, heated up to \( T_{0} \) degrees, is brought into a room, the temperature in which is constant and equal to \( \alpha \) degrees.

8.1.4.10 Geometric and physical applications

1.48 $

Problem: The decay rate of radium is proportional to its initial amount. During a year \( 0,44 \mathrm{mg} \). decays from each gram of radium. How many years will it take for the half of the radium to decay?

8.1.4.11 Geometric and physical applications

1.73 $

Problem: The amount of light, absorbed when passing through a thin layer of water, is proportional to the amount of incident light and the thickness of the layer. Knowing that when passing through a 2 -meter-thick layer of water \( 1 / 3 \) of the initial flow of light is absorbed, find what portion of it will reach a depth of 12 meters. clip2net.com

8.1.4.12 Geometric and physical applications

1.73 $

Problem: Find the integral curve of the equation \( y \cdot y^{\prime} \). \( y^{\prime \prime}=\left(y^{\prime}\right)^{3}+\left(y^{\prime \prime}\right)^{2} \), tangent to the line \( x+y=0 \) at the origin.

8.1.4.13 Geometric and physical applications

3.7 $

Problem: A particle of mass \( m \) moves linearly under the action of a constant force \( F \). Find the velocity of the movement of the particle and the distance, as a function of time, if at the initial moment both of them are equal to zero, and the resistance of the environment is proportional to the square of the velocity.

8.1.4.14 Geometric and physical applications

3.21 $

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