You will explore the characteristics of ballistic motion in cases where a body collides with and is reflected elastically from massive obstacles during flight. Air resistance can be neglected in this problem. Acceleration due to gravity is $g=9.8~{\rm m/s}^2$.

The following mathematical facts may be useful. Consider the polynomial of the second degree $f(x)=ax^2+bx+c$, $a<0$.

1. The graph of this function is a parabola, the $x$ coordinate of the vertex of which is $x=-\frac{b}{2a}$. In other words $f(x)$ takes on a maximum value in this point.
2. Consider equation $f(x)=0$. Let it have two roots, $x_1$ and $x_2$. Then the following equalities are valid:
   \begin{equation*}
   x_1+x_2=-\frac{b}{a},\quad x_1x_2=\frac{c}{a}.
   \end{equation*}

Part A. Reflection from the wall (1,5 points)

Consider the collision of a body with a massive vertical wall. We will work in a Cartesian coordinate system such that the $Oy$ axis is vertical. Let the velocity of the body in the instant before the collision be $\vec{v}_0=(v_{0x}, v_{0y})$. The wall moves with velocity $\vec{u}=(-u,0)$, $u>0$.

A1 Find relative body velocity $\vec{v_0}'=(v_{0x}', v_{0y}')$ in wall frame. Express the coordinates of the vector in terms of $v_{0x}$, $v_{0y}$, and $u$.

A S

Since the obstacle is massive and the impact is perfectly elastic, in the frame of reference fixed to the wall, the magnitudes of the body's velocity before and after the collision are identical, and the angle of reflection is equal to the angle of incidence. This assumption must be maintained throughout the problem.

A2 Find the velocity $\vec{v}'=(v_{x}', v_{y}')$ of the body in the wall's frame of reference immediately after reflection from the wall. Express the vector's coordinates in terms of $v_{0x}'$, $v_{0y}'$.

A S

A3 Find the velocity $\vec{v}=(v_{x}, v_{y})$ of the body in the laboratory frame of reference immediately after reflection from the wall. Express the coordinates of the vector in terms of $v_{0x}$, $v_{0y}$, $u$.

A S

Part B. Flight with one reflection (5 points)

Now let's assume that the $Oy$ axis is directed vertically upwards, and the $Ox$ axis is horizontal and lies in the plane of the throw. The body is located at the origin of the coordinate system $x=y=0$. At the initial moment of time ($t=0$), the body is thrown with initial velocity $\vec{v_{0}}=(v_{0x}, v_{0y})$.

B1 Determine the coordinates of the body $x(t)$, $y(t)$ at the moment $t$ after launch. Express the answer in terms of $v_{0x}$, $v_{0y}$, $g$, $t$.

A S

B2 It is known that at some point in time the body was at a height of $h$ above the earth's surface. Write down an equation that allows you to find the time $t$ after launch, when this could have happened. Express the equation coefficients using $v_{0y}$, $g$, $h$.

A S

It is known that at times $t_1$ and $t_2$ after the throw, the body was at the same height above the earth's surface.

B3 Determine this height and find the vertical projection $v_{0y}$ of the initial velocity. Express the anwser using $t_1$, $t_2$, $g$.

S

B4 Determine the total flight time. Express it using $t_1$, $t_2$.

S

B5 Determine the maximum height of the trajectory. Express it using $t_1$, $t_2$, $g$.

A S

There is a massive vertical wall at the point where the body in question should land. Simultaneously with the launch of the body, it begins to move towards it.

B6 Express horizontal projection $v_x$ of body speed right after reflection from wall using $v_{0x}$ and magnitude of wall velocity $u$.

A S

B7 Let a wall collide with a body at time $\tau$. Express the value of the wall velocity $u$ using $v_{0x}$, $\tau$ and the flight time $t$ without the wall.

A S

Two experiments were conducted, as described in the problem statement. Their difference lay solely in the initial velocity of the wall. It is known that in the first case, the collision occurred at time $t_1$, and in the second - at time $t_2$, but in both cases at the same height. The distance between the places where the body falls in the first and second experiments is equal to $L$.

B8 Define $v_{0x}$. Express the anwser using $L$, $t_1$, $t_2$.

A S

B9 Calculate the numerical value of the magnitude $v_0$ of the initial velocity in case of $L=16~{\rm m}$, $t_1=2~{\rm s}$, $t_2=3~{\rm s}$.

A S

B10 Calculate the numerical value of the initial distance $S$ between the wall and the body's throw point.

A S

Part C. Bouncing between two walls (3,5 points)

Let us now consider the launch of the body from the starting point with a certain fixed initial velocity $v_0$, but a free launch angle $\alpha$. A safety parabola is the boundary of an area beyond which a projectile cannot hit, regardless of the angle of launch $\alpha$. First you need to get the equation of the safety parabola.

Let again axis $Oy$ is vertical, axis $Ox$ is horizontal and lies in the plane of the throw. The body is located at the origin of the coordinate system $x=y=0$.

C1 Define maximum height $h$, which can be reached by a body thrown with an initial velocity of $v_0$. Express the anwser using $v_0$, $g$.

A S

C2 Determine the maximum throw range $L$. Express the anwser using $v_0$, $g$.

A S

Assuming that the boundary of the region is a parabola, we can conclude that its equation has the form $y=ax^2+bx+c$.

C3 Using the previous results, determine the coefficients $a$, $b$ and $c$. Express the anwsers using $v_0$, $g$.

A S

The boy throws a small ball from a height of $H$. The initial velocity is $v_0$.

C4 Determine the maximum horizontal throw distance $R$ of the ball by the time it hits the surface. Express the anwser using $v_0$, $g$, $H$.

A S

Now let the person perform the same throw, being strictly in the middle between two vertical walls, the distance between which is $l$.

C5 What is the maximum number of bounces of the ball from both walls by the time it lands on the ground? Express your answer in terms of $v_0$, $g$, $H$, $l$.

A S

Hint. Imagine that there is a mirror hanging on the wall against which the body is hitting. What will be the movement of the body after the bounce in the mirror?

C6 Calculate the numerical value in case of $v_0=5~{\rm m/s}$, $H=4~{\rm m}$, $l=2~{\rm m}$.

A S