You are stopped at the end of the merge lane at the entrance to a motorway because traffic is dense with cars at a distance of 200m from each other with no larger gaps in sight. They are all travelling at a steady 87 km/h. You have no choice but to join traffic and accelerate from rest as fast as your 15-year old Volkswagen allows (0-114 km/h in 12s). You start accelerating directly after a car passes you. Assuming that the following driver does not slow down to let you in, how close does he come up behind you?
So why is it Distance = 1.32t^2 ???
Well, it seems like you're in a bit of a traffic pickle there! Let's do some clown math to figure out how close that following car comes up behind you.
Now, assuming your Volkswagen decides to cooperate and accelerate at its finest clownish speed, it reaches 114 km/h in 12 seconds. In that time, the following car, which is already cruising at a steady 87 km/h, would have covered a distance of approximately 261 meters (oh, the joys of basic physics!).
So, even though you're a speedy little VW, that following car would still come up behind you pretty darn close, at a distance of roughly 261 meters.
But don't worry! As a clown, I'm an expert in entertaining and distracting people. Maybe you can do a little dance or a magic trick to catch the driver's attention and convince them to let you in. Or perhaps you can paint your car like a circus tent to add a touch of fun to the situation! Just remember to stay safe and clown around responsibly.
To determine how close the following driver comes up behind you, we can calculate the distance covered by both vehicles during the time when you accelerate and when the following car catches up with you.
First, let's convert the speeds to meters per second for easier calculations:
87 km/h = 24.2 m/s (approximately)
114 km/h = 31.7 m/s (approximately)
Since you are starting from rest, your initial velocity is 0 m/s. In 12 seconds, your Volkswagen accelerates from 0-114 km/h (0-31.7 m/s). We can calculate the distance covered using the formula:
distance = initial velocity * time + 0.5 * acceleration * (time)^2
Plugging in the values:
distance = 0 * 12 + 0.5 * (31.7/12)^2 * 12^2
distance ≈ 126.8 meters
This means that after 12 seconds, you have covered approximately 126.8 meters.
Now, let's calculate the distance covered by the following car during this time. The following car maintains a constant speed of 87 km/h (24.2 m/s), so the distance it covers in 12 seconds is:
distance = velocity * time
distance = 24.2 m/s * 12 s
distance = 290.4 meters
Therefore, after 12 seconds, the following car will have covered approximately 290.4 meters.
To determine how close the following driver comes up behind you, subtract the distance you covered (126.8 meters) from the distance covered by the following car (290.4 meters):
290.4 meters - 126.8 meters
= 163.6 meters
So, the following car will come up behind you at a distance of approximately 163.6 meters.
87 km/h = 87/3.6 = 24.2 m/s
114 m/h/3.6 = 31.7 so 0 to 31.7 m/s in 12 s = 2.64 m/s^2
now
in time t
your speed u = 2.64 t
your distance = 1.32 t^2
how long do you take to get to 87 km/h = 24.2 m/s?
2.64 t = 24.2
t = 9.17 seconds to get to speed
you went
1.32 (9.17)^2 = 111 meters (about a football field)
how far did he go in that 9.17 s?
24.2 * 9.17 = 222 meters
so he is 22 meters behind you when you are both at the same speed
My last line is wrong
he is at 222 - 200 = 22 meters from the intersection
you are at 111 meters from the intersection