<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<meta property="og:title" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<title>A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. </title>
$1.1.2.$ A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle $\alpha_1 = 44^{\circ}$, distance $R_1 = 100\;km$. At a time interval of $5\;s$ after this moment, the coordinates of the airplane on the radar: angle $\alpha_2 = 46^{\circ}$, distance $R_2 = 100\;km$. In a Cartesian coordinate system with the $y$-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction.
</p>
<h4>Solution</h4>
<p>
$O_1$ is the initial position of the airplane. $O_2$ is the final position of the airplane. In time $t$ the airplane will fly the distance:
$$S = v_0 \cdot t$$
We find the distance $S$ from the isosceles triangle $AO_1O_2$, where the angle $O_1AO_2 = 2^{\circ}$. Then
$$S = 2R \cdot sin1^{\circ}$$
Where $R_1=R_2=R$
$$v_0 \cdot t = 2R \cdot sin1^{\circ}$$
Desired speed
$$v_0 = \frac{2R \cdot sin1^{\circ}}{t}$$
At small angle $sin\alpha\approx\alpha$ expressed in radians, i.e. $1^{\circ} = \frac{\pi}{180}$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<meta name="description" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<meta property="og:title" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<meta property="og:title" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<meta property="og:description" content="A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. ">
<title>A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. </title>
<title>A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle \alpha_1 = 44^{\circ}, distance R_1 = 100\;km. At a time interval of 5\;s after this moment, the coordinates of the airplane on the radar: angle \alpha_2 = 46^{\circ}, distance R_2 = 100\;km. In a Cartesian coordinate system with the y-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction. </title>
$1.1.2.$ A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle $\alpha_1 = 44^{\circ}$, distance $R_1 = 100\;km$. At a time interval of $5\;s$ after this moment, the coordinates of the airplane on the radar: angle $\alpha_2 = 46^{\circ}$, distance $R_2 = 100\;km$. In a Cartesian coordinate system with the $y$-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction.
$1.1.2.$ A radar determines the coordinates of a flying airplane by measuring the angle between the direction to the North Pole and the direction to the airplane and the distance from the radar to the airplane. At some point in time, the position of the airplane was determined by the coordinates: angle $\alpha_1 = 44^{\circ}$, distance $R_1 = 100\;km$. At a time interval of $5\;s$ after this moment, the coordinates of the airplane on the radar: angle $\alpha_2 = 46^{\circ}$, distance $R_2 = 100\;km$. In a Cartesian coordinate system with the $y$-axis pointing north and with radar at the origin, represent the position of the airplane at both moments of time; determine the modulus and direction of its velocity. Count the angle in a clockwise direction.
</p>
</p>
<h4>Solution</h4>
<h4>Solution</h4>
<p>
<p>
$O_1$ is the initial position of the airplane. $O_2$ is the final position of the airplane. In time $t$ the airplane will fly the distance:
$O_1$ is the initial position of the airplane. $O_2$ is the final position of the airplane. In time $t$ the airplane will fly the distance:
$$S = v_0 \cdot t$$
$$S = v_0 \cdot t$$
We find the distance $S$ from the isosceles triangle $AO_1O_2$, where the angle $O_1AO_2 = 2^{\circ}$. Then
We find the distance $S$ from the isosceles triangle $AO_1O_2$, where the angle $O_1AO_2 = 2^{\circ}$. Then
$$S = 2R \cdot sin1^{\circ}$$
$$S = 2R \cdot sin1^{\circ}$$
Where $R_1=R_2=R$
Where $R_1=R_2=R$
$$v_0 \cdot t = 2R \cdot sin1^{\circ}$$
$$v_0 \cdot t = 2R \cdot sin1^{\circ}$$
Desired speed
Desired speed
$$v_0 = \frac{2R \cdot sin1^{\circ}}{t}$$
$$v_0 = \frac{2R \cdot sin1^{\circ}}{t}$$
At small angle $sin\alpha\approx\alpha$ expressed in radians, i.e. $1^{\circ} = \frac{\pi}{180}$
At small angle $sin\alpha\approx\alpha$ expressed in radians, i.e. $1^{\circ} = \frac{\pi}{180}$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
