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Question Number 83606 by MJS last updated on 04/Mar/20
a car drives at a speed of 120 km/hr it starts to brake at a road mark A and passes a road mark B at a speed of 60 km/hr. acceleration is constant. the distance AB is 4 km. (1) calculate the acceleration (2) calculate the time between A and B (3) there are n reflector posts between A and B. calculate the speed of the car at each of them (find a function for the speed depending on the distance traveled)
$$\mathrm{a}\:\mathrm{car}\:\mathrm{drives}\:\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{120}\:\mathrm{km}/\mathrm{hr} \\ $$$$\mathrm{it}\:\mathrm{starts}\:\mathrm{to}\:\mathrm{brake}\:\mathrm{at}\:\mathrm{a}\:\mathrm{road}\:\mathrm{mark}\:{A}\:\mathrm{and} \\ $$$$\mathrm{passes}\:\mathrm{a}\:\mathrm{road}\:\mathrm{mark}\:{B}\:\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{60}\:\mathrm{km}/\mathrm{hr}.\:\mathrm{acceleration}\:\mathrm{is}\:\mathrm{constant}.\:\mathrm{the} \\ $$$$\mathrm{distance}\:{AB}\:\mathrm{is}\:\mathrm{4}\:\mathrm{km}. \\ $$$$\left(\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{acceleration} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{time}\:\mathrm{between}\:{A}\:\mathrm{and}\:{B} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{there}\:\mathrm{are}\:{n}\:\mathrm{reflector}\:\mathrm{posts}\:\mathrm{between}\:{A} \\ $$$$\:\:\:\:\:\:\:\mathrm{and}\:{B}.\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}\:\mathrm{at} \\ $$$$\:\:\:\:\:\:\:\mathrm{each}\:\mathrm{of}\:\mathrm{them}\:\left(\mathrm{find}\:\mathrm{a}\:\mathrm{function}\:\mathrm{for}\:\mathrm{the}\right. \\ $$$$\left.\:\:\:\:\:\:\:\mathrm{speed}\:\mathrm{depending}\:\mathrm{on}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{traveled}\right) \\ $$
Answered by mr W last updated on 04/Mar/20
v_A =120 km/h v_B =60 km/h (1) a=((v_B ^2 −v_A ^2 )/(2s))=((60^2 −120^2 )/(2×4))=−1350 km/h^2 =−(5/(48)) m/s^2 (2) t=((v_B −v_A )/a)=((60−120)/(−1350))=(2/(45)) h=160 s (3) v_((i)) =speed at i−th reflector ((v_((i)) ^2 −v_A ^2 )/(2s_((i)) ))=a v_((i)) =(√(v_A ^2 +2as_((i)) ))=v_A (√(1−2×((1350)/(120^2 ))×(i/(n+1))×4)) v_((i)) =120(√(1−((0.75i)/(n+1)))) (km/h)
$${v}_{{A}} =\mathrm{120}\:{km}/{h} \\ $$$${v}_{{B}} =\mathrm{60}\:{km}/{h} \\ $$$$\left(\mathrm{1}\right) \\ $$$${a}=\frac{{v}_{{B}} ^{\mathrm{2}} −{v}_{{A}} ^{\mathrm{2}} }{\mathrm{2}{s}}=\frac{\mathrm{60}^{\mathrm{2}} −\mathrm{120}^{\mathrm{2}} }{\mathrm{2}×\mathrm{4}}=−\mathrm{1350}\:{km}/{h}^{\mathrm{2}} =−\frac{\mathrm{5}}{\mathrm{48}}\:{m}/{s}^{\mathrm{2}} \\ $$$$\left(\mathrm{2}\right) \\ $$$${t}=\frac{{v}_{{B}} −{v}_{{A}} }{{a}}=\frac{\mathrm{60}−\mathrm{120}}{−\mathrm{1350}}=\frac{\mathrm{2}}{\mathrm{45}}\:{h}=\mathrm{160}\:{s} \\ $$$$\left(\mathrm{3}\right) \\ $$$${v}_{\left({i}\right)} ={speed}\:{at}\:{i}−{th}\:{reflector} \\ $$$$\frac{{v}_{\left({i}\right)} ^{\mathrm{2}} −{v}_{{A}} ^{\mathrm{2}} }{\mathrm{2}{s}_{\left({i}\right)} }={a} \\ $$$${v}_{\left({i}\right)} =\sqrt{{v}_{{A}} ^{\mathrm{2}} +\mathrm{2}{as}_{\left({i}\right)} }={v}_{{A}} \sqrt{\mathrm{1}−\mathrm{2}×\frac{\mathrm{1350}}{\mathrm{120}^{\mathrm{2}} }×\frac{{i}}{{n}+\mathrm{1}}×\mathrm{4}} \\ $$$${v}_{\left({i}\right)} =\mathrm{120}\sqrt{\mathrm{1}−\frac{\mathrm{0}.\mathrm{75}{i}}{{n}+\mathrm{1}}}\:\left({km}/{h}\right) \\ $$

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