Two automobiles A and B are approaching each other in...

Two automobiles $A$ and $B$ are approaching each other in adjacent highway lanes. At $t=0, A$ and $B$ are $3200 \mathrm{ft}$ apart, their speeds are $v_{A}=65 \mathrm{mi} / \mathrm{h}$ and $v_{B}=40 \mathrm{mi} / \mathrm{h},$ and they are at points $P$ and $Q,$ respectively. Knowing that $A$ passes point $Q 40$ s after $B$ was there and that $B$ passes point $P 42$ s after $A$ was there, determine $(a)$ the uniform accelerations of $A$ and $B,(b)$ when the vehicles pass each other, $(c)$ the speed of $B$ at that time.

Key Concepts

Uniform Acceleration

This concept refers to motion in which an object's acceleration remains constant throughout its motion. In problems of uniform acceleration, the changes in velocity occur at a steady rate, making it possible to use specific kinematic equations to describe the motion. The assumption of constant acceleration simplifies the analysis and allows for the derivation of relationships between distance, velocity, acceleration, and time.

Kinematic Equations

Kinematic equations are the fundamental formulas used to relate displacement, velocity, acceleration, and time in problems involving constant acceleration. These equations, such as s = ut + (1/2)at² and v = u + at, are essential tools for solving motion problems, as they provide a systematic method to determine unknown variables when certain initial conditions and constraints are known.

Relative Motion Analysis

Relative motion analysis involves examining the motion of two or more objects with respect to each other. In scenarios where two objects are moving in opposite directions or along parallel paths, analyzing the problem from the perspective of one object can simplify the calculation of meeting points, speeds, and distances. This technique is especially useful when the movements are interdependent or when calculating the effect of different start times or delays.

Time-Distance Relationships

Time-distance relationships are concerned with the connection between the time taken to travel a certain distance and the speed of the moving object. This concept is critical in determining how delays or differences in departure times affect the meeting point of two moving vehicles. Understanding these relationships helps in setting up the equations needed to compute unknown quantities based on known time intervals and distances.

Unit Conversion

Unit conversion is the process of converting quantities from one unit of measure to another to ensure consistency and correctness in calculations. In physics problems, it is often necessary to convert speeds, distances, or time values into compatible units, such as converting miles per hour to feet per second, before applying the kinematic equations. This step is crucial for arriving at accurate and meaningful results.

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