Question

Two cars start off to race with velocities v1 and v2 and travel in a straight line with uniform acceleration a1 and a2 respectively. If the race ends in a dead heat ( i.e they reach the finishing point at the same time) Prove the length of the course is

2(v1-v2)(v1a2-v2a2)/(a1-a2)^2

Answer 1

Question #77629, Physics / Molecular Physics | Thermodynamics

Two cars start off to race with velocities v1 and v2 and travel in a straight line with uniform acceleration a1 and a2 respectively. If the race ends in a dead heat (i.e they reach the finishing point at the same time) Prove the length of the course is

$$2(v1-v2)(v1a2-v2a1)/(a1-a2)^2$$

Solution

$$d = v_1 t + \frac{a_1}{2} t^2 = v_2 t + \frac{a_2}{2} t^2$$ $$(v_1 - v_2) t = \frac{1}{2} t^2 (a_2 - a_1)$$ $$t = \frac{2(v_1 - v_2)}{a_2 - a_1}$$

Thus,

$$d = v_2 \left(\frac{2(v_1 - v_2)}{a_2 - a_1}\right) + \frac{a_2}{2} \left(\frac{2(v_1 - v_2)}{a_2 - a_1}\right)^2 = \frac{2 v_2 (v_1 - v_2) (a_2 - a_1) + 2 a_2 (v_1 - v_2)^2}{(a_2 - a_1)^2}$$ $$d = \frac{2(v_1 - v_2)(v_1 a_2 - v_2 a_1)}{(a_1 - a_2)^2}$$

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