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Acceleration and time taken by a car to increase speed (8329)
Jueves 10 de octubre de 2024, por
A car, which has uniformly accelerated motion, increases its speed from 18 km/h to 72 km/h over a straight distance of 37.5 meters. Calculate the time taken for this journey and its acceleration.
To make the problem homogeneous, the first step is to express the speeds in SI units:
![\left 18\ \dfrac{\cancel{km}}{\cancel{h}}\cdot \dfrac{10^3\ m}{1\ \cancel{km}}\cdot \dfrac{1\ \cancel{h}}{3.6\cdot 10^3\ s} = {\color[RGB]{0,112,192}{\bm{5\ m\cdot s^{-1}}}} \atop 72\ \dfrac{\cancel{km}}{\cancel{h}}\cdot \dfrac{10^3\ m}{1\ \cancel{km}}\cdot \dfrac{1\ \cancel{h}}{3.6\cdot 10^3\ s} = {\color[RGB]{0,112,192}{\bm{20\ m\cdot s^{-1}}}} \right \}](local/cache-vignettes/L398xH105/67c9d605de0bbcc641dfbf99e22bae3c-78c83.png?1732959459 "\left 18\ \dfrac{\cancel{km}}{\cancel{h}}\cdot \dfrac{10^3\ m}{1\ \cancel{km}}\cdot \dfrac{1\ \cancel{h}}{3.6\cdot 10^3\ s} = {\color[RGB]{0,112,192}{\bm{5\ m\cdot s^{-1}}}} \atop 72\ \dfrac{\cancel{km}}{\cancel{h}}\cdot \dfrac{10^3\ m}{1\ \cancel{km}}\cdot \dfrac{1\ \cancel{h}}{3.6\cdot 10^3\ s} = {\color[RGB]{0,112,192}{\bm{20\ m\cdot s^{-1}}}} \right \}")
You can relate the change in speed and the distance covered with the car’s acceleration:
![v_f^2 = v_i^2 + 2ad\ \to\ \color[RGB]{2,112,20}{\bm{a = \frac{(v_f^2 - v_i^2)}{2d}}}](local/cache-vignettes/L317xH53/a55b922add5dd0650cc307c0d373c97c-f5ae0.png?1732959459 "v_f^2 = v_i^2 + 2ad\ \to\ \color[RGB]{2,112,20}{\bm{a = \frac{(v_f^2 - v_i^2)}{2d}}}")
Substitute the values and calculate:
![a = \frac{(20^2 - 5^2)\ m\cancel{^2}\cdot s^{-2}}{2\cdot 37.5\ \cancel{m}} = \fbox{\color[RGB]{192,0,0}{\bm{5\ m\cdot s^{-2}}}}](local/cache-vignettes/L350xH51/c1facad935205b9ca3ac232a004e0209-c22c0.png?1732959459 "a = \frac{(20^2 - 5^2)\ m\cancel{^2}\cdot s^{-2}}{2\cdot 37.5\ \cancel{m}} = \fbox{\color[RGB]{192,0,0}{\bm{5\ m\cdot s^{-2}}}}")
The time needed to make this speed change is:
![v_f = v_i + a\cdot t\ \to\ \color[RGB]{2,112,20}{\bm{t = \frac{v_f - v_i}{a}}}](local/cache-vignettes/L289xH44/14fb46f95265619345e66352a4f43496-7e9b7.png?1732959459 "v_f = v_i + a\cdot t\ \to\ \color[RGB]{2,112,20}{\bm{t = \frac{v_f - v_i}{a}}}"){kind=small}
Substitute and calculate:
![t = \frac{(20 - 5)\ \cancel{m}\cdot \cancel{s^{-1}}}{5\ m\cdot s^{\cancel{-2}}} = \fbox{\color[RGB]{192,0,0}{\bf 3\ s}}](local/cache-vignettes/L257xH48/deca9ca7581aa024e04fec4a7e96bc1f-39cf2.png?1732959459 "t = \frac{(20 - 5)\ \cancel{m}\cdot \cancel{s^{-1}}}{5\ m\cdot s^{\cancel{-2}}} = \fbox{\color[RGB]{192,0,0}{\bf 3\ s}}"){kind=small}