if 100m sprinters accelerate from rest for 3.5s at 2.8m/s^2, how far have they run to this point? how long will it take them to complete the 100m sprint, assuming they maintain their speed the rest of the way?
For the first part use s = ut + (1/2)at2, where s = distance, u = initial speed, a = acceleration, t = time:
$${\mathtt{s}} = {\mathtt{0}}{\mathtt{\,\times\,}}{\mathtt{3.5}}{\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}{\mathtt{2.8}}{\mathtt{\,\times\,}}{{\mathtt{3.5}}}^{{\mathtt{2}}} \Rightarrow {\mathtt{s}} = {\mathtt{17.15}}$$
So distance = 17.15m
For the second part find the speed from v = u + at:
$${\mathtt{v}} = {\mathtt{0}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2.8}}{\mathtt{\,\times\,}}{\mathtt{3.5}} \Rightarrow {\mathtt{v}} = {\mathtt{9.8}}$$
v = 9.8m/s
At this speed the sprinter will cover the remaining distance in time t2 = (100 - 17.15)/9.8 seconds
$${\mathtt{t2}} = {\frac{\left({\mathtt{100}}{\mathtt{\,-\,}}{\mathtt{17.15}}\right)}{{\mathtt{9.8}}}} \Rightarrow {\mathtt{t2}} = {\mathtt{8.454\: \!081\: \!632\: \!653\: \!061\: \!2}}$$
t2 ≈ 8.45 s
So, total time for the 100m is 3.5 + 8.45 seconds = 11.95 seconds
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For the first part use s = ut + (1/2)at2, where s = distance, u = initial speed, a = acceleration, t = time:
$${\mathtt{s}} = {\mathtt{0}}{\mathtt{\,\times\,}}{\mathtt{3.5}}{\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}{\mathtt{2.8}}{\mathtt{\,\times\,}}{{\mathtt{3.5}}}^{{\mathtt{2}}} \Rightarrow {\mathtt{s}} = {\mathtt{17.15}}$$
So distance = 17.15m
For the second part find the speed from v = u + at:
$${\mathtt{v}} = {\mathtt{0}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2.8}}{\mathtt{\,\times\,}}{\mathtt{3.5}} \Rightarrow {\mathtt{v}} = {\mathtt{9.8}}$$
v = 9.8m/s
At this speed the sprinter will cover the remaining distance in time t2 = (100 - 17.15)/9.8 seconds
$${\mathtt{t2}} = {\frac{\left({\mathtt{100}}{\mathtt{\,-\,}}{\mathtt{17.15}}\right)}{{\mathtt{9.8}}}} \Rightarrow {\mathtt{t2}} = {\mathtt{8.454\: \!081\: \!632\: \!653\: \!061\: \!2}}$$
t2 ≈ 8.45 s
So, total time for the 100m is 3.5 + 8.45 seconds = 11.95 seconds
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