Question

three balls A,B and C have 10 g, 20 g and 30g mass respectively. The ball A is released from rest and ball B and C are thrown straight upward with initial speed of 5 m /s and 10 m/s upward respectively. the motions of all masses are fall

Answer 1

Task:

Three balls A, B and C have $10\mathrm{g}$ , $20\mathrm{g}$ and $30\mathrm{g}$ mass respectively. The ball A is released from rest and ball B and C are thrown straight upward with initial speed of $5\mathrm{m/s}$ and $10\mathrm{m/s}$ upward respectively. What are the motions of all masses?

Solution:

$$A: s _ {A} = s _ {A _ {0}} + v _ {A _ {0}} \cdot t + \frac {a \cdot t ^ {2}}{2} = s _ {0} - \frac {g \cdot t ^ {2}}{2}$$ $$B: s _ {B} = s _ {B _ {0}} + v _ {B _ {0}} \cdot t + \frac {a \cdot t ^ {2}}{2} = 5 \frac {m}{s} \cdot t - \frac {g \cdot t ^ {2}}{2}$$ $$C: s _ {C} = s _ {C _ {0}} + v _ {C _ {0}} \cdot t + \frac {a \cdot t ^ {2}}{2} = 1 0 \frac {m}{s} \cdot t - \frac {g \cdot t ^ {2}}{2}$$

Motion of the center of mass:

$$\begin{array}{l} s _ {A B C} = \frac {\sum_ {i = 1} ^ {n} m _ {i} s _ {i}}{\sum_ {i = 1} ^ {n} m _ {i}} = \frac {m _ {A} s _ {A} + m _ {B} s _ {B} + m _ {C} s _ {C}}{m _ {A} + m _ {B} + m _ {C}} = \\ = \frac {0 . 0 1 k g \cdot \left(s _ {A _ {0}} - \frac {g \cdot t ^ {2}}{2}\right) + 0 . 0 2 k g \cdot \left(5 \frac {m}{s} \cdot t - \frac {g \cdot t ^ {2}}{2}\right) + 0 . 0 3 k g \cdot \left(1 0 \frac {m}{s} \cdot t - \frac {g \cdot t ^ {2}}{2}\right)}{0 . 0 6 k g} = \\ = \frac {s _ {A _ {0}}}{6} - \frac {g \cdot t ^ {2}}{1 2} + \frac {1 0 m}{6 s} \cdot t - \frac {2 \cdot g \cdot t ^ {2}}{1 2} + \frac {3 0 m}{6 s} \cdot t - \frac {3 \cdot g \cdot t ^ {2}}{1 2} = \frac {s _ {0}}{6} + \frac {4 0 m}{6 s} \cdot t - \frac {6 \cdot g \cdot t ^ {2}}{1 2} = \\ = \frac {s _ {A _ {0}}}{6} + 6 \frac {2}{3} \left(\frac {m}{s}\right) \cdot t - \frac {g \cdot t ^ {2}}{2} \\ \end{array}$$

Answer:

$$s _ {A B C} = \frac {s _ {A _ {0}}}{6} + 6 \frac {2}{3} \left(\frac {m}{s}\right) \cdot t - \frac {g \cdot t ^ {2}}{2}$$