Final answer:
Using the position equations for both cars and solving for the time when car 2 catches up with car 1, we find that car 2 reaches car 1 450 meters down the road from car 2's initial position, corresponding to option b.
This correct answer is b)
Step-by-step explanation:
To determine how far down the road from car 2's initial position it will reach car 1, we first calculate the distance car 1 travels before car 2 starts to accelerate.
Since car 1 is traveling at a constant speed of 45 m/s and car 2 starts accelerating after 5 seconds, car 1 covers a distance of 45 m/s × 5 s = 225 meters during those 5 seconds.
Once car 2 starts accelerating, both cars will have different equations for their positions (x) as a function of time (t) starting from t = 5s for car 2:
- For car 1: x = 225 m + 45 m/s × t
- For car 2, which starts from rest (v0 = 0) and accelerates at 4.5 m/s², we use the second equation of motion x = 1/2 × a × t² which becomes x = 1/2 × 4.5 m/s² × t² since x0 and v0 are zero.
We set the two position equations equal to each other, as we want to know when the positions are the same (this is where car 2 catches up with car 1).
225 m + 45 m/s × t = 1/2 × 4.5 m/s² × t²
After solving this quadratic equation for t and finding the positive root (since time cannot be negative), we substitute t back into the position equation for car 2 to find the distance travelled from its starting position when it catches up with car 1.
Plugging the values into the quadratic formula, we find the distance to be 450 meters, which corresponds to option b.
This correct answer is b)