Answer to Question #129903 in Physics for Jack man

Question #129903

A car travels 1km it starts to accelerate at 2.5m/s it attains a velocity of 12.5m/s the car continues at this velocity for some time and decelerates at 3m/s^2 until it stops. What is the total time for the whole distance?


1
Expert's answer
2020-08-20T09:48:47-0400

Let the first acceleration be a1=2.5m/s2a_1 = 2.5 m/s^2 and the second one a2=3m/s2a_2 = 3m/s^2. The Initial speed is v0=0v_0 = 0, the next one is v1=12.5m/sv_1 = 12.5m/s and the final is v2=0v_2 = 0. The total distance is d=1km=1000md = 1km = 1000m.

Let's find the time t1t_1 of the first acceleration from v0v_0 to v1v_1

By definition, the acceleration is:


a1=v1v0t1t1=v1v0a1=12.502.5=5sa_1 = \dfrac{v_1 - v_0}{t_1}\\ t_1 = \dfrac{v_1 - v_0}{a_1} = \dfrac{12.5-0}{2.5} = 5s

During this time the car covered the distance:


d1=a1t122=2.5522=31.25md_1 = \dfrac{a_1t_1^2}{2} = \dfrac{2.5\cdot 5^2}{2} = 31.25 m

Let's find the time t3t_3 of the last acceleration. Again, by definition:


a2=v2v1t3t1=v2v1a2=012.53=256sa_2 = \dfrac{v_2-v_1}{t_3}\\ t_1 = \dfrac{|v_2 - v_1|}{a_2} = \dfrac{|0-12.5|}{3} = \dfrac{25}{6}s

During this time the car covered the distance:


d3=a2t322=3252262=156.25md_3 = \dfrac{a_2t_3^2}{2} = \dfrac{3\cdot 25^2}{2\cdot 6^2} = 156.25 m

The distance in the middle, covered with a constant speed v1v_1 is then:


d2=dd1d3=1000m31.25m156.25m=812.5md_2 = d-d_1 - d_3 = 1000m - 31.25m - 156.25m = 812.5m

Thus, the time during which it was covered is:


t2=d2/v1=65st_2 = d_2/v_1 = 65 s

Thus, the total time is:


t=t1+t2+t3=5s+256s+65s74.17st = t_1 + t_2 + t_3 = 5s + \dfrac{25}{6}s + 65s \approx 74.17s

Answer. 74.17s.



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