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202202071437 Dynamics Figures (Elementary) Q3

1.17. A stationary object explodes into two fragments of relative mass 1:100. At the instant of break-up, the larger mass has a velocity of 10\,\mathrm{m\, s^{-1}}. Calculate i. the velocity of the smaller mass, ii. the ratio of their kinetic energies at this instant.

Extracted from M. Nelkon. (1971). Graded Exercises and Worked Examples in Physics.

Solution.

Provided that u=0, m_1:m_2=1:100, and v_2=10\,\mathrm{m\, s^{-1}}.

By the law of conservation of linear momentum,

\begin{aligned} M\mathbf{u} & = m_1\mathbf{v}_1+m_2\mathbf{v}_2 \\ (M)(0\,\hat{\mathbf{i}}) & = \bigg(\frac{1}{101}M\bigg) (-v_1\,\hat{\mathbf{i}})+\bigg(\frac{100}{101}M\bigg) (+10\,\hat{\mathbf{i}}) \\ v_1 & = 1000 \\ \mathbf{v}_1 & = -1000\,\hat{\mathbf{i}}\\ \end{aligned}

The ratio of their kinetic energies is given by

\begin{aligned} &\quad \textrm{KE}_1:\textrm{KE}_2 \\ & = \frac{\frac{1}{2}m_1v_1^2}{\frac{1}{2}m_2v_2^2} \\ & = \frac{(\frac{1}{2})(\frac{1}{101}M)(1000)^2}{(\frac{1}{2})(\frac{100}{101}M)(10)^2} \\ & = 100:1 \\ \end{aligned}