Question

the red car’s front bumper is one car - length behind a blue car’s rear bumper, while a truck (which is three car lengths long) approaches in the other lane. to the drivers of these vehicles, the separation of the front bumper of the blue car and the front bumper of the truck appears to be about twenty car - lengths. all three vehicles are traveling at the speed limit, and the blue car and truck maintain a constant speed at all times. the driver of the red car is considering the possibility of accelerating to pass the blue car.
the speed limit of the road is $13.3\frac{m}{s}$, and the driver of the red car decides to pass. the driver of the blue car is unaware that she is passing, and does not change his speed. if a “car length” is 4.49 meters, find the minimum acceleration (in $\frac{m}{s^2}$) the red car must have to successfully pass. provide at least two decimal places

Explanation:

Step1: Calculate total distance to cover

First, find the total distance the red car needs to travel to safely pass. It needs to cover: 1 car length (initial gap) + 1 car length (its own length) + 20 car lengths (blue-truck gap) + 3 car lengths (truck length). Total car lengths = $1+1+20+3=25$.
Distance $d = 25 \times 4.49 = 112.25$ meters.

Step2: Find time until truck meets blue car

Both blue car and truck move toward each other at speed $v=13.3\ \frac{m}{s}$. Closing speed is $13.3 + 13.3 = 26.6\ \frac{m}{s}$. The initial gap between blue car and truck is 20 car lengths: $20 \times 4.49 = 89.8$ meters.
Time $t = \frac{\text{gap}}{\text{closing speed}} = \frac{89.8}{26.6} = 3.37594$ seconds.

Step3: Use kinematic equation for red car

The red car's motion is described by $d = v_0 t + \frac{1}{2} a t^2$, where $v_0=13.3\ \frac{m}{s}$, $d=112.25$ m, $t=3.37594$ s. Rearrange to solve for $a$:
$a = \frac{2(d - v_0 t)}{t^2}$

Substitute values:
$v_0 t = 13.3 \times 3.37594 = 44.900$
$d - v_0 t = 112.25 - 44.900 = 67.35$
$t^2 = (3.37594)^2 = 11.397$
$a = \frac{2 \times 67.35}{11.397} = \frac{134.7}{11.397} \approx 11.82$

Answer:

$11.82\ \frac{m}{s^2}$