Question

Two cars start off to race with velocities $ 4m/s $ and $ 2m/s $ and travel in a straight line with uniform acceleration $ 1m/{s^2} $ and $ 2m/{s^2} $ respectively. If they reach the final point at the same time, then the length of the path is
A) 24m
B) 30m
C) 12m
D) 45m

Answer 1

Hint: In this solution, we will calculate the distance travelled by the two cars given with their initial velocities and acceleration. Since they reach the final point at the same time, they will cover the same amount of distance in equal amounts of time.

Formula used In this solution, we will use the following formula:
-Second equation of kinematics: $ d = ut + \dfrac{1}{2}a{t^2} $ where $ d $ is the distance covered by an object travelling with an initial velocity $ u $ , constant acceleration $ a $ in time $ t $

Complete step by step answer:
We’ve been given that two cars start off to race with different velocities and different acceleration and they reach the finish line at the same time.
For the first car, we have $ {u_1} = 4\,m/s $ and $ {a_1} = 1\,m/{s^2} $ , so we can use the second equation of kinematics and write
 $ d = 4t + \dfrac{1}{2}(1){t^2} $
For the first car, we have $ {u_2} = 2\,m/s $ and $ {a_2} = 2\,m/{s^2} $ , so we can again use the second equation of kinematics. Since both the cars cross the finish point at the same point, they will cover an equal distance $ d $ in time $ t $ , so we can write
 $ d = 2t + \dfrac{1}{2}(2){t^2} $
 $ \Rightarrow d = 2t + {t^2} $
Comparing equation (1) and (2), we can write
 $ 4t + \dfrac{{{t^2}}}{2} = 2t + {t^2} $
Which can be simplified to
 $ {t^2} - 4t = 0 $
Or
 $ t(t - 4) = 0 $
Hence $ t = 0 $ or $ t = 4 $ seconds. Since $ t = 0 $ is arbitrary, the time taken by the cars to cross the finish point will be $ t = 4\,s $ . The distance the first car will cover at this time will be
 $ d = 4(4) + \dfrac{1}{2}(1){(4)^2} $
 $ \Rightarrow d = 24m $
Hence option (A) will be the correct choice.

Note:
The distance travelled by both the cars will be the same so we can only calculate the distance travelled by one of the cars to find the answer. This scenario is only possible when the car with the greater initial velocity has a smaller acceleration than the second car.