Pho.rs: Movement with bounces

A1 ^0.50 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 $u$.

1 \[\vec{v}_0' = (v_{0x}+u,v_{0y})\]	0.50
2 Wrong sign before $u$	-0.30

A2 ^0.50 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 $v_{0x}'$, $v_{0y}'$.

1 Expression from A1 is correctly used, e.g.
\[\vec{v}'=(-v_{0x}-u,v_{0y})\]

0.50

A3 ^0.50 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 $v_{0x}$, $v_{0y}$, $u$.

1 Expression from A2 is correctly used, e.g. \[\vec{v}=(-v_{0x}-2u,v_{0y})\]	0.50
2 \[\vec{v}=(-v_{0x},v_{0y})\]	0.30

B1 ^0.40 Determine the coordinates of the body $x(t)$, $y(t)$ at the moment $t$ after launch. Express the answer in terms of $v_{0x}$, $v_{0y}$, $g$, $t$.

1 Equation of motion with constant acceleration is used \[x = x_0 + v_{x0}t+\frac{a_xt^2}{2} \]	0.20
2 $a_x=0$	0.10
3 $a_y=-g$	0.10

B2 ^0.40 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 $v_{0y}$, $g$, $h$.

1 Expression from B1 is used, e.g.: \[v_{0y}t-\frac{gt^2}{2} = h\]	0.20
2 \[t_{1,2}=\frac{v_{0y}\pm\sqrt{v_{0y}^2-2gh}}{g}\]	0.20

B3 ^1.00 Determine this height and find the vertical projection $v_{0y}$ of the initial velocity. Express the anwser using $t_1$, $t_2$, $g$.

1 Expression from B1 is used, e.g.: \[v_{0y}t - \frac{gt^2}{2}=H\]	0.20
2 Coefficient is correctly expressed, e.g. \[A=\frac{g}{2}\]	0.10
3 Coefficient is correctly expressed, e.g \[B=-v_{0y}\]	0.10
4 Coefficient is correctly expressed \[C=H\]	0.10
5 Application of properties of roots (sum and product)	2 × 0.15
6 \[v_{0y}=\frac{g(t_1+t_2)}{2}\]	0.10
7 \[H=\frac{gt_1t_2}{2}\]	0.10

B4 ^0.40 Determine the total flight time. Express it using $t_1$, $t_2$.

1 M1 Expression from B1 is used, e.g.: \[v_{0y}t-\frac{gt^2}{2}=0\]	0.10
2 M1 \[T=\frac{2v_{0y}}{g}\]	0.20
3 M2 Using symmetry	0.30
4 \[T=t_1+t_2\]	0.10

B5 ^0.60 Determine the maximum height of the trajectory. Express it using $t_1$, $t_2$, $g$.

1 \[t_\mathrm{vert}=\frac{T}{2}\]	0.20
2 $t=T/2$ is substituted to expression from B1, e.g. \[y_\mathrm{max}=v_{0y} \frac{T}{2} - \frac{gT^2}{8}\]	0.20
3 \[y_\mathrm{max}=\frac{g(t_1+t_2)^2}{8}\]	0.20

B6 ^0.40 Express horizontal projection $v_x$ of body speed right after reflection from wall using $v_{0x}$ and magnitude of wall velocity $u$.

1 Expression from A3 is used, e.g.
\[v'_x = - v_{0x}-2u\]

0.40

B7 ^0.40 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.

1 Initial position of wall is expressed \[X=v_{0x}T\]	0.10
2 \[X = v_{0x} \tau + u \tau \]	0.20
3 \[u = v_{0x} \frac{T - \tau}{\tau}\]	0.10

B8 ^1.00 Define $v_{0x}$. Express the anwser using $L$, $t_1$, $t_2$.

1 Expression from B7 is used: \[u_i=v_{0x} \frac{T - t_i}{t_i}\]	0.40
2 Expression from B6 is used: \[v_x' = - v_{0x} - 2u\]	0.20
3 \[ X_1 = v_{0x} \frac{t_1^2 - t_1 t_2 - 2t_2^2}{t_1}\]	0.10
4 \[ X_2 = v_{0x} \frac{t_2^2 - t_1 t_2 - 2t_1^2}{t_2}\]	0.10
5 \[X_1 - X_2 = v_{0x} \frac{2(t_1 + t_2)^2(t_1-t_2)}{t_1t_2}\]	0.10
6 \[v_{0x} = \frac{Lt_1 t_2}{2(t_1+t_2)^2(t_2-t_1)}\]	0.10

B9 ^0.20 Calculate the numerical value of the magnitude $v_0$ of the initial velocity in case of $L=16~{\rm m}$, $t_1=2~{\rm s}$, $t_2=3~{\rm s}$.

1 Expression from B8 is used	0.10
2 \[v_{0x}=25.1~\mathrm{m}/\mathrm{s}\]	0.10

B10 ^0.20 Calculate the numerical value of the initial distance $S$ between the wall and the body's throw point.

1 \[S=X=v_{0x}T\]	0.10
2 \[S = 63~\mathrm{m}\]	0.10

C1 ^0.50 Define maximum height $h$, which can be reached by a body thrown with an initial velocity of $v_0$. Express the anwser using $v_0$, $g$.

1 \[h_\mathrm{max}=\frac{v_0^2}{2g}\]

0.50

C2 ^0.50 Determine the maximum throw range $L$. Express the anwser using $v_0$, $g$.

1 \[L=\frac{2v_0^2 \sin \alpha \cos \alpha}{g}\]	0.30
2 \[L=\frac{v_0^2}{g}\]	0.20

C3 ^0.70 Using the previous results, determine the coefficients $a$, $b$ and $c$. Express the anwsers using $v_0$, $g$.

1 Safety parabolla is derived or $h$ and $L$ are correctly used	0.40
2 \[a=-\frac{g}{2v_0^2}\]	0.10
3 \[b=0\]	0.10
4 \[c=\frac{v_0^2}{2g}\]	0.10

C4 ^0.80 Determine the maximum horizontal throw distance $R$ of the ball by the time it hits the surface. Express the anwser using $v_0$, $g$, $H$.

1 Equation of safety parabolla is used	0.60
2 Direct derivation is used but it containts some mistakes	0.30
3 \[R=\frac{v_0 \sqrt{v_0^2 + 2gH}}{g}\]	0.20

C5 ^0.50 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 $v_0$, $g$, $H$, $l$.

1 Geometrical idea that $R$ could be used to determine $N$	0.30
2 Correct expression without rounding: \[Nl - \frac{l}{2} = R\]	0.10
3 \[N = \left\lfloor \frac{v_0 \sqrt{v_0^2 + 2gH}}{g} - \frac{1}{2} \right\rfloor \]	0.10

C6 ^0.50 Calculate the numerical value in case of $v_0=5~{\rm m/s}$, $H=4~{\rm m}$, $l=2~{\rm m}$.

1 \[N=2\]

0.50

Movement with bounces

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