Car Motion Problem: Acceleration, Velocity & Distance

Hey everyone! Let's break down this physics problem step-by-step. We've got a car starting from rest, covering 64 meters in 8 seconds. Our mission is to figure out its acceleration, velocity after those 8 seconds, and the distance it travels in the next 16 seconds. Buckle up, let's dive in!

1. Calculating Acceleration

To calculate the acceleration of the car, we'll use one of the fundamental equations of motion. Since the car starts from rest, its initial velocity is zero. We know the distance traveled (64 meters) and the time taken (8 seconds). The equation that connects these variables is: d = v₀t + (1/2)at²

Where:

• d is the distance traveled
• v₀ is the initial velocity
• t is the time elapsed
• a is the acceleration

In our case, d = 64 meters, v₀ = 0 m/s, and t = 8 seconds. Plugging these values into the equation, we get:

64 = (0)(8) + (1/2)a(8)² 64 = 0 + (1/2)a(64) 64 = 32a

Now, we solve for 'a' by dividing both sides of the equation by 32:

a = 64 / 32 a = 2 m/s²

So, the acceleration of the car is 2 meters per second squared. This means the car's velocity increases by 2 meters per second every second. Understanding acceleration is crucial as it tells us how quickly the velocity of the car is changing. A higher acceleration means a faster change in velocity, while a lower acceleration means a slower change. In this scenario, the car steadily gains speed, making it easier to predict its future motion. Acceleration is not just a theoretical concept; it directly impacts the real-world driving experience, affecting everything from braking distances to how quickly you can merge onto a highway. This calculation sets the foundation for understanding the rest of the car's motion in this problem.

2. Determining the Velocity After 8 Seconds

Now that we know the car's acceleration, let's figure out its velocity after 8 seconds. We can use another equation of motion for this:

v = v₀ + at

Where:

• v is the final velocity
• v₀ is the initial velocity
• a is the acceleration
• t is the time elapsed

We already know v₀ = 0 m/s, a = 2 m/s², and t = 8 seconds. Plugging these values into the equation:

v = 0 + (2)(8) v = 16 m/s

Therefore, the car's velocity after 8 seconds is 16 meters per second. This is a significant value as it gives us a snapshot of how fast the car is moving at a specific point in time. Knowing the velocity is essential for understanding the car's momentum and kinetic energy, which are crucial for analyzing collisions or other dynamic scenarios. The car's velocity not only represents its instantaneous speed but also provides a basis for predicting its future position if it were to maintain this speed. In the context of driving, understanding the relationship between acceleration and velocity is paramount for making safe and efficient maneuvers, such as overtaking or merging onto highways. This calculation further enhances our understanding of the car's motion, preparing us to tackle the next part of the problem.

3. Calculating the Distance Traveled in the Next 16 Seconds

Finally, let's calculate the distance the car travels in the next 16 seconds. Here, things get a bit interesting because the car is no longer starting from rest. It already has a velocity of 16 m/s. So, we'll use the same equation we used in the first step, but with a non-zero initial velocity:

d = v₀t + (1/2)at²

This time, v₀ = 16 m/s, a = 2 m/s², and t = 16 seconds. Plugging these values in:

d = (16)(16) + (1/2)(2)(16)² d = 256 + (1)(256) d = 256 + 256 d = 512 meters

So, the car travels 512 meters in the next 16 seconds. This distance is considerably larger than the initial 64 meters, highlighting the impact of the car's constant acceleration over time. Calculating the distance covered in a given time interval is a fundamental skill in physics, with applications ranging from vehicle navigation to projectile motion analysis. In this context, understanding how the car's initial velocity and acceleration combine to determine its displacement is crucial. This final calculation provides a comprehensive picture of the car's motion throughout the entire time frame, solidifying our understanding of its kinematics.

Summary

To sum it all up, we've determined the car's acceleration to be 2 m/s², its velocity after 8 seconds to be 16 m/s, and the distance it travels in the next 16 seconds to be 512 meters. By applying the equations of motion step-by-step, we've successfully solved this problem. Understanding these concepts is super helpful for analyzing real-world scenarios involving motion. Keep practicing, and you'll become a physics whiz in no time!