To solve the problem of when and where the car catches up with the 18-wheeler, we can break it down into a few logical steps. We'll use the equations of motion for the car, which is accelerating, and the constant speed of the 18-wheeler. Let's analyze the situation step by step.
Understanding the Motion of Both Vehicles
The car starts from rest and accelerates at a constant rate, while the 18-wheeler moves at a constant speed. We can use the following equations of motion:
- For the car, which starts from rest: d = 0.5 * a * t^2
- For the 18-wheeler, which moves at a constant speed: d = v * t
Defining Variables
Let's define the variables:
- a = acceleration of the car = 2 m/s²
- v = speed of the 18-wheeler = 11 m/s
- t = time in seconds
- d_c = distance traveled by the car
- d_t = distance traveled by the 18-wheeler
Setting Up the Equations
At the moment the traffic light turns green, the car begins to accelerate, while the 18-wheeler has already passed it. We need to find the time when both vehicles have traveled the same distance from the traffic light.
For the car:
d_c = 0.5 * a * t^2 = 0.5 * 2 * t^2 = t^2
For the 18-wheeler:
d_t = v * t = 11 * t
Finding the Catch-Up Point
To find when the car catches up to the 18-wheeler, we set the distances equal to each other:
t^2 = 11 * t
Rearranging this gives us:
t^2 - 11t = 0
Factoring out t:
t(t - 11) = 0
This gives us two solutions: t = 0 (the starting point) and t = 11 seconds.
Calculating the Distance
Now that we know it takes 11 seconds for the car to catch up, we can find the distance traveled by either vehicle during this time. Let's use the 18-wheeler's distance:
d_t = v * t = 11 * 11 = 121 meters
Summary of Results
In summary, the car will catch up with the 18-wheeler:
- After 11 seconds
- At a distance of 121 meters beyond the traffic light
This analysis shows how the concepts of acceleration and constant speed interact in a real-world scenario, illustrating the importance of understanding motion in physics.
