91Ó°ÊÓ

Kinematic equations allow us to analyze the movement of objects with constant acceleration. They help us find unknown variables like initial velocity, final velocity, acceleration, distance, and time.
They are particularly useful in solving problems related to motion in one dimension, like a car skidding to a stop on a road.
In this exercise, we use the kinematic equation:\[ v^2 = v_0^2 + 2a d \]Here, \(v\) is the final velocity (which is 0 because the car stops), \(v_0\) is the initial velocity, \(a\) is acceleration, and \(d\) is the distance traveled while skidding. By rearranging the equation to \(v_0^2 = -2a d\), we can solve for \(v_0\), the initial speed of the car when the driver hit the brakes.
Understanding how to manipulate these equations is crucial.

• Set the final velocity to zero when an object stops.
• Use negative acceleration (deceleration) when calculating a stopping scenario.
• Solve for the variable of interest by rearranging the equation.

Knowing this makes tackling such physics problems clearer and more manageable.

The coefficient of friction is a value that represents the frictional force between two surfaces. There are two main types: static friction, which keeps an object at rest, and kinetic friction, which acts on objects in motion.
In this scenario, we're concerned with kinetic friction because the car's wheels are sliding along the road.
The coefficient of kinetic friction is usually denoted \(\mu_k\) and is a dimensionless number. In this example, it is given as 0.750.
This value, alongside gravity, is used to calculate the force of friction, which allows us to determine deceleration:\[ a = \mu_k \cdot g \]where \(g\) is the acceleration due to gravity, approximately \(32.2 \ \text{ft/s}^2\) in these calculations.
This simple formula helps calculate how quickly the car would stop, based on friction alone. So, a higher coefficient of friction causes more rapid deceleration.
Here are key points:

• The coefficient of friction depends on both surface materials.
• A higher value indicates greater frictional force.
• Kinetic friction plays a crucial role in stopping vehicles.

Understanding the coefficient's role is vital in problems involving frictional force.

Velocity conversion is important to express speed in different units. It ensures the speed is understood in context, such as changing from feet per second (ft/s) to miles per hour (mi/h).
In the exercise, the car's skidding speed was originally found in \(\text{ft/s}\).
To convert it into \(\text{mi/h}\), we use the conversion factor:\[ 1 \ \text{mi/h} = 1.46667 \ \text{ft/s} \]This factor comes from converting both distances (miles and feet) and time units (hours and seconds).
By multiplying the speed in \(\text{ft/s}\) by the conversion factor \[ v_0 \times \frac{1 \ \text{mi/h}}{1.46667 \ \text{ft/s}} \]we obtain the speed in \(\text{mi/h}\). Thus, understanding conversion helps analyze speeds better in appropriate terms for legal and communicational standards.
Keep these points in mind:

• Check initial and final units to ensure accurate conversion.
• Multiply by the conversion factor, not divide, to convert units correctly.
• Knowing common conversion rates like \(1 \ \text{mi/h} = 1.46667 \ \text{ft/s}\) simplifies calculations.