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Velocity and acceleration

Задачи
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Разбалловка

0.0
Сумма баллов
A1  0.50 Express $\left|\vec v\right|^2$ in terms of $v_x$ and $v_y$.

1 $$\left|\vec v\right|^2 = v_x^2+v_y^2$$ 0.50
A2  0.50 Express $\left|\vec a\right|$ in terms of $\left|\vec a_\parallel\right|$ and $\left|\vec a_\perp\right|$.

1 $$\left|\vec a\right| = \sqrt{\left|\vec a_\parallel\right|^2+\left|\vec a_\perp\right|^2}$$ 0.50
A3  1.00 Prove that $\left(\vec v \cdot \vec a\right) = \left(\vec v\cdot \vec a_\parallel\right)$ and $\left|\left(\vec v \cdot \vec a\right)\right| = \left|\vec v\right|\cdot \left|\vec a_\parallel\right|$.

1 Доказано, что $\left(\vec v \cdot \vec a\right) = \left(\vec v\cdot \vec a_\parallel\right)$. 0.50
2 Доказано, что $\left|\left(\vec v \cdot \vec a\right) \right|= \left|\vec v\right|\cdot \left|\vec a_\parallel\right|$. 0.50
A4  1.00 Express $\left|\vec a_\perp\right|$ in terms of $\left|\vec a\right|$, $\left|\vec v\right|$ and $\left(\vec v \cdot \vec a\right)$.

1 $$\left|\vec a_\parallel\right| = \frac{\left|\left(\vec v \cdot \vec a\right) \right|}{\left|\vec v\right|}$$ 0.50
2 $$\left|\vec a_\perp\right| = \sqrt{\left|\vec a\right|^2 - \frac{\left(\vec v \cdot \vec a\right) ^2}{\left|\vec v\right|^2}}$$ 0.50
A5  0.50 Prove that
$$\frac{d\left(f(t)\right)^2}{dt} = Af(t)\dot f(t),$$where $A$ is a constant, and find the value of $A$. If you are unable to complete this step, assume $A = 2$.

1 Доказано и получено:
$$A=2$$ 		0.50
A6  1.00 Using the results of A1 and A5, express $\frac{d\left(\vec v\cdot\vec v\right)}{dt}$ in terms of $\left(\vec v\cdot\vec a\right)$.

1 $$\frac{d\left(\vec v\cdot \vec v\right)}{dt} = 2\left(\vec v \cdot \vec a\right)$$ 1.00
B1  1.00 Express $v_x$ and $v_y$ in terms of $v$ and $\varphi$.

1 $$v_x = v\cos\varphi$$$$v_y = v\sin\varphi$$ 2 × 0.50
B2  1.00 Express $\frac{d\left(\vec v\cdot\vec v\right)}{dt}$ in terms of $v$ and $\dot v$.

1 $$\frac{d\left(\vec v\cdot \vec v\right)}{dt} = 2v\dot v$$ 1.00
B3  1.00 Express $a_x$ and $a_y$ in terms of $v$, $\varphi$, $\dot v$ and $\dot\varphi$.

1 $$a_x = \dot v \cos\varphi - v\sin\varphi\dot \varphi$$$$a_y = \dot v \sin\varphi + v\cos\varphi\dot\varphi$$ 2 × 0.50
B4  1.00 Express $\left|\vec a_\perp\right|$ in terms of $v$, $\varphi$, $\dot v$ and $\left|\dot\varphi\right|$.

1 $$\left|\vec a_\perp\right| = v\left|\dot\varphi\right|$$ 1.00
B6  0.50 Express $\dot v$ in terms of $v$ and $\left(\vec v\cdot\vec a\right)$.

1 $$\dot v = \frac{\left(\vec v\cdot\vec a\right)}{v}$$ 0.50
B7  1.00 Express $\left|\dot\varphi\right|$ in terms of $v$, $\left(\vec v\cdot\vec a\right)$ and $\left|\vec a\right|$.

1 $$\left|\dot\varphi\right| = \frac{1}{v}\sqrt{\left|\vec a\right|^2 - \frac{\left(\vec v \cdot \vec a\right) ^2}{v^2}}$$ 1.00