Answer:
<em>Answer: Option d.</em>
Explanation:
<u>Accelerated Motion
</u>
When an object changes its spped in the same amounts in the same times, the acceleration is constant and its value is
Where
are the final speed, initial speed, and time taken to change them, respectively
From the above equation we can know
The distance traveled is computed as
The question talks about a car moving in a straight line with constant acceleration. It goes through the points p,q,r such as
The ratio of the distances traveled in each segment is
being
the distance from p to q and
the distance from q to r
It means that
From the equation for speed
Replacing [1] into [2]
Solving for a
![\displaystyle a=\frac{v_r-v_p}{t_1+t_2}\ \ \ .........[3]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20a%3D%5Cfrac%7Bv_r-v_p%7D%7Bt_1%2Bt_2%7D%5C%20%5C%20%5C%20.........%5B3%5D)
We now write the equation for both distances
.
![\displaystyle x_1=v_pt_1+\frac{at_1^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x_1%3Dv_pt_1%2B%5Cfrac%7Bat_1%5E2%7D%7B2%7D)
![\displaystyle x_2=v_qt_2+\frac{at_2^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x_2%3Dv_qt_2%2B%5Cfrac%7Bat_2%5E2%7D%7B2%7D)
Using [1] again
![\displaystyle x_2=(v_p+at_1)t_2+\frac{at_2^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x_2%3D%28v_p%2Bat_1%29t_2%2B%5Cfrac%7Bat_2%5E2%7D%7B2%7D)
Since
![x_2=2x_1](https://tex.z-dn.net/?f=x_2%3D2x_1)
We have
![\displaystyle (v_p+at_1)t_2+\frac{at_2^2}{2}=2\left (v_pt_1+\frac{at_1^2}{2}\right )](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%28v_p%2Bat_1%29t_2%2B%5Cfrac%7Bat_2%5E2%7D%7B2%7D%3D2%5Cleft%20%28v_pt_1%2B%5Cfrac%7Bat_1%5E2%7D%7B2%7D%5Cright%20%29)
Operating
![\displaystyle v_pt_2+at_1t_2+\frac{at_2^2}{2}=2v_pt_1+at_1^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20v_pt_2%2Bat_1t_2%2B%5Cfrac%7Bat_2%5E2%7D%7B2%7D%3D2v_pt_1%2Bat_1%5E2)
Rearranging
![\displaystyle v_pt_2-2v_pt_1=at_1^2-at_1t_2-\frac{at_2^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20v_pt_2-2v_pt_1%3Dat_1%5E2-at_1t_2-%5Cfrac%7Bat_2%5E2%7D%7B2%7D)
Factoring both sides
![\displaystyle v_p(t_2-2t_1)=\frac{a}{2}\left (2t_1^2-2t_1t_2-t_2^2 \right )](https://tex.z-dn.net/?f=%5Cdisplaystyle%20v_p%28t_2-2t_1%29%3D%5Cfrac%7Ba%7D%7B2%7D%5Cleft%20%282t_1%5E2-2t_1t_2-t_2%5E2%20%20%5Cright%20%29)
Replacing the equation [3] for a
:
![\displaystyle 2v_p(t_2-2t_1)=\frac{v_r-v_p}{t_1+t_2}\left (2t_1^2-2t_1t_2-t_2^2 \right )](https://tex.z-dn.net/?f=%5Cdisplaystyle%202v_p%28t_2-2t_1%29%3D%5Cfrac%7Bv_r-v_p%7D%7Bt_1%2Bt_2%7D%5Cleft%20%282t_1%5E2-2t_1t_2-t_2%5E2%20%20%5Cright%20%29)
Replacing
, and operating the denominator
![\displaystyle 10(t_2-2t_1)\left (t_1+t_2 \right )=20\left (2t_1^2-2t_1t_2-t_2^2 \right )](https://tex.z-dn.net/?f=%5Cdisplaystyle%2010%28t_2-2t_1%29%5Cleft%20%28t_1%2Bt_2%20%20%5Cright%20%29%3D20%5Cleft%20%282t_1%5E2-2t_1t_2-t_2%5E2%20%20%5Cright%20%29)
Operating and simplifying, we get a second-degree equation
![\displaystyle t_2^2+t_1t_2-2t_1^2=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20t_2%5E2%2Bt_1t_2-2t_1%5E2%3D0)
Factoring
![(t_2-t_1)(t_2+2t_1)=0](https://tex.z-dn.net/?f=%28t_2-t_1%29%28t_2%2B2t_1%29%3D0)
The only positive and valid answer is
![t_2=t_1](https://tex.z-dn.net/?f=t_2%3Dt_1)
Or equivalently
![\displaystyle \frac{t_1}{t_2}=1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bt_1%7D%7Bt_2%7D%3D1)
The option d. is correct