Problem 29 The brakes on your automobile ar... [FREE SOLUTION] (2024)

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In physics, kinematics is the branch that deals with the motion of objects without considering the forces that cause the motion. When analyzing the movement of a car decelerating, we need to apply kinematic equations. These equations relate the initial and final velocities, acceleration (or deceleration), time, and displacement. Here’s a breakdown of key terms:

• Velocity (\textbf{v}): The rate at which an object changes its position. Measured in meters per second (m/s).
• Acceleration (\textbf{a}): The rate at which an object changes its velocity. When a car slows down, this is termed deceleration and usually has a negative value. Measured in meters per second squared (m/s\textsuperscript{2}).
• Time (\textbf{t}): The duration over which the change occurs. Measured in seconds (s).

We use these components in the formula:\[ a = \frac{v_f - v_i}{t} \]Rearranging it to find time (t) gives:\[ t = \frac{v_f - v_i}{a} \]Applying these principles helps in determining how quickly a car can decelerate to a lower speed.

When solving physics problems, it’s crucial to ensure all units are consistent. In the given problem, the speeds are provided in kilometers per hour (km/h), but we need them in meters per second (m/s). This is done using the conversion factor: 1 km/h = 0.27778 m/s. Here’s a detailed conversion:
Initial speed (137 km/h) conversion:

• Multiply by the conversion factor: 137 * 0.27778 = 38.06 m/s.

Final speed (90 km/h) conversion:

• Multiply by the conversion factor: 90 * 0.27778 = 25 m/s.

These conversions are essential for accurately applying the kinematic formulas, which require consistent units of measure. Converting units ensures that you don’t end up with incorrect values due to mismatched units.

Graphing is a vital part of physics as it provides visual insight into the motion described by the kinematic equations. For this problem, we consider two types of graphs: velocity (\textbf{v}) vs. time (\textbf{t}) and displacement (\textbf{x}) vs. time (t):

• **Velocity vs. Time:** This graph for deceleration is a straight line with a negative slope. Starting from 38.06 m/s at t = 0 and ending at 25 m/s at t = 2.51 s. The slope represents the deceleration rate.


• **Distance vs. Time:** To plot this, we use the formula for displacement:\[ x = v_i * t + 0.5 * a * t^2 \]Plugging in the values:
  \[ x = 38.06 * t + 0.5 * (-5.2) * t^2 \]This gives a parabolic curve, starting at the origin (0,0) and curving upwards before slowing down due to deceleration.

Graphing these equations helps in understanding how the vehicle's speed decreases over time and how far it travels during deceleration. Visual tools like these are very helpful in making abstract concepts more concrete.