91Ó°ÊÓ

Uniform acceleration occurs when an object's velocity changes at a constant rate over time. In other words, the object's speed increases or decreases evenly during its motion. This is a fundamental concept in kinematics, the branch of physics that describes the motion of objects.

In the given problem, the car accelerates at a uniform rate of 0.600 \( \mathrm{m/s}^2 \). This means the car's speed increases by 0.600 meters per second every second while it attempts to pass the truck. This uniform acceleration allows us to use kinematic equations to predict the car's velocity and distance covered at any given time.

Understanding uniform acceleration helps us analyze how quickly an object can change its speed, which is crucial for calculating how long it will take to complete a particular journey. For example, with a known starting velocity and constant acceleration, we can determine exactly when the car will be ahead of the truck by its required distance, as shown in the problem.

Kinematic equations are essential tools in physics that describe the motion of an object under uniform acceleration. They allow us to calculate unknown variables such as time, distance, and final velocity when others are known. These equations are:

• \( v_f = v_i + a \cdot t \)
• \( d = v_i \cdot t + \frac{1}{2} a \cdot t^2 \)
• \( v_f^2 = v_i^2 + 2a \cdot d \)

In the exercise, the problem asks us to determine how long it takes for the car to pass the truck (time \( t \)), the distance traveled during this time (\( d \)), and the car's final speed when it has done so (\( v_f \)).

To solve these, we start with the second kinematic equation to find the time by plugging in the known initial velocity \( v_i = 20 \text{ m/s} \), acceleration \( a = 0.600 \text{ m/s}^2 \), and total passing distance \( d = 75.5 \text{ meters} \). This equation rearranges into a quadratic form that helps pinpoint the exact amount of time needed to complete the action, which is\[ 75.5 = 20t + 0.3t^2 \].

By solving this equation, we can find how long the car must accelerate before safely passing the truck.

The quadratic formula is a powerful mathematical tool used to solve equations of the form \( ax^2 + bx + c = 0 \). In physics, particularly in kinematics, it's often applied to find time, distance, or velocity when an object is under uniform acceleration.

The solution for the quadratic equation is given by:

• \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In our case, the quadratic equation derived from the kinematic equation is \( 0.3t^2 + 20t - 75.5 = 0 \). The coefficients are \( a = 0.3 \), \( b = 20 \), and \( c = -75.5 \). Plugging these into the quadratic formula, we calculate the discriminant \( b^2 - 4ac \), which helps determine if real solutions exist for time \( t \).

The formula provides two solutions, but only the positive one is physically meaningful for time, hence resulting in \( t \approx 3.54 \text{ seconds} \). Understanding the quadratic formula and its application ensures we can solve for unknowns in complex kinematics problems, providing insights into the dynamics of motion.