Solution: A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them……..
A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them…..
Understanding the Scenario: A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them…..m/s2
The situation involves a car and a van at a traffic light. Both vehicles are stationary at the start, with the car positioned in front of the van. Importantly, there is an initial 1-meter gap between the two vehicles.
Once the traffic light turns green, the car begins to move forward. It starts accelerating, and after a certain period of time, it reaches a constant speed. The van, however, does not start moving immediately. Instead, it waits until the gap between the car and van increases from the initial 1 meter to 4 meters. At this point, the van begins to accelerate in the same manner as the car.
2. Acceleration of the Car: A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them……
The car’s motion consists of two distinct phases:
a) Acceleration Phase
In the first phase, the car accelerates from rest, gradually increasing its speed. This acceleration continues until the car reaches its desired final speed. During this period, the car’s velocity is not constant but increases steadily over time.
b) Constant Speed Phase
After accelerating for a certain amount of time, the car reaches its target speed of 15 meters per second (m/s). Once this speed is achieved, the car continues to move forward at this constant velocity. The transition between these two phases is crucial, as the van has yet to start moving during this time.
3. Initial Gap Between the Car and Van:A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them……
At the start, the gap between the car and the van is 1 meter. The van remains stationary while the car starts to move forward, meaning this gap begins to widen as the car accelerates. The critical moment occurs when the gap increases to 4 meters.
The question arises: why does the van start moving only when the gap reaches 4 meters? This is because the van must allow the car enough space to begin its motion, avoiding an immediate collision. It’s important to recognize that the van’s driver is aware of this and waits for the car to gain enough distance before initiating their own acceleration.
4. Van’s Delayed Start: A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them…..
a) Why the Van Waits
The van waits until there is a sufficient distance between itself and the car before moving. This delay ensures that the vehicles do not come too close to each other, which could cause an accident. From the moment the light turns green, the van’s driver monitors the gap between the two vehicles. Initially, the gap is 1 meter, but as the car accelerates, this distance increases.
b) When the Van Starts Moving
Once the gap reaches 4 meters—that is, an increase of 3 meters from the initial gap—the van begins its own acceleration. At this point, the car has already been in motion for a while and is in the midst of its acceleration phase. The key detail here is that the van does not start moving until the car has created a safe distance.
5. The Van’s Acceleration: A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them……
Once the van starts moving, it accelerates in the same way the car did. The acceleration of both vehicles is identical, meaning the van also takes some time to reach the same final speed as the car—15 m/s.
a) Van’s Acceleration Phase
Like the car, the van goes through an acceleration phase. Starting from rest, it gradually increases its speed until it too reaches 15 m/s. However, because the van starts accelerating later than the car, it will naturally take longer to catch up to the car’s current position.
b) Reaching the Final Speed
The van reaches the same final speed of 15 m/s, but this happens later than the car since it started accelerating 2 seconds after the car. Despite this delay, both vehicles eventually achieve the same speed, albeit at different times.
6. Movement After Reaching Constant Speed
Both the car and the van eventually reach their final speed of 15 m/s. At this point, neither vehicle is accelerating anymore, and they are both moving forward at a constant velocity. Even though they are traveling at the same speed, the gap between them will remain constant from this point onward.
a) Car’s Motion at Constant Speed
Once the car reaches its final speed of 15 m/s, it maintains this speed indefinitely. During this phase, the car no longer accelerates, meaning its speed does not change over time. The distance the car travels during this phase is simply a product of its constant speed and the time that has passed.
b) Van’s Motion at Constant Speed
The van, similarly, reaches the same constant speed of 15 m/s, but this occurs after the car has been traveling at that speed for a couple of seconds. Once the van reaches 15 m/s, it also maintains this speed indefinitely, just like the car. At this point, the relative motion between the two vehicles stabilizes because both are traveling at the same velocity.
7. Gap Between the Vehicles When Both Reach 15 m/s
Once both the car and the van are moving at 15 m/s, the gap between them remains constant. However, the question is: what is this final gap?
To understand the final gap, it’s important to recall that the van started its motion 2 seconds after the car. During these 2 seconds, the car was accelerating while the van was stationary. This initial delay plays a crucial role in determining the final distance between the two vehicles.
a) Distance Covered by the Car During the Van’s Delay
During the first 2 seconds, the car was accelerating. This means it was steadily increasing its speed while the van remained stationary. The gap between the two vehicles continued to grow during this period. Once the van started accelerating, it began to close the distance somewhat, but by the time it reached the same speed as the car, the gap had already widened significantly.
b) The Constant Gap at 15 m/s
By the time both vehicles reach their final speed, the gap between them is fixed. Since both are now traveling at the same speed, the relative distance between them will not change. This gap is a result of the van’s delayed start and the car’s motion during the van’s stationary period. As a result, the final gap is larger than the initial 1-meter gap, but it remains constant as long as both vehicles continue moving at 15 m/s.
8. Conceptual Overview of the Solution: A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them…..
Let’s summarize the conceptual approach to understanding this problem:
- Initial Condition: The car and van start with a 1-meter gap between them at the traffic light.
- Car’s Acceleration: The car begins to accelerate when the light turns green, moving forward and increasing the gap.
- Van’s Delayed Start: The van waits until the gap between it and the car increases to 4 meters before starting to accelerate.
- Van’s Acceleration: The van accelerates in the same way as the car but starts 2 seconds later.
- Constant Speed Phase: Both vehicles eventually reach a constant speed of 15 m/s. At this point, the gap between them stops changing.
- Final Gap: The final gap between the car and van is the result of the van’s delayed start and the distance the car covered during this delay.
9. Key Insights and Principles
This problem illustrates several important concepts in mechanics, particularly the relationship between distance, speed, and time in the context of motion with constant acceleration. It demonstrates how an initial difference in the timing of motion—such as the van starting 2 seconds later—can lead to a fixed separation between two objects, even when both eventually move at the same speed.
a) Role of Acceleration
The acceleration phase is critical because it determines how quickly the vehicles reach their final speed. Both the car and the van go through this phase, but the timing of when they enter and exit it plays a key role in determining the gap between them.
b) Constant Speed and Fixed Gap
Once both vehicles reach the same constant speed, the distance between them stabilizes. This is because, when two objects move at the same velocity, the relative distance between them remains unchanged unless one of them accelerates again.
c) Time and Relative Motion
The difference in when the car and the van start moving leads to the final gap between them. The longer the van waits to start moving, the larger this gap will be. This demonstrates the importance of timing in understanding the dynamics of motion.
Conclusion A car is waiting at traffic lights with a van behind it. There is a 1 metre gap between them…..
In this problem, the car and the van exhibit similar motion patterns but with a crucial difference in timing. The delayed start of the van leads to a fixed separation between the two vehicles, despite the fact that both eventually travel at the same speed. This example highlights fundamental principles of motion, such as acceleration, constant velocity, and the relationship between distance, speed, and time.
Understanding these principles conceptually allows us to appreciate the mechanics of the situation without needing to delve into detailed calculations. The solution reveals how small variations in timing and motion can lead to significant outcomes in terms of distance and relative positioning.
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