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 $v_{A}=65 \mathrm{mil} / \mathrm{h}$ and $v_{B}=40 \mathrm{mi} / \mathrm{h},$ and they are at points $P$ and $Q$, respectively. Knowing that $A$ passes point $Q 40 \mathrm{s}$ 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

Solving Simultaneous Equations

Solving simultaneous equations is a mathematical technique used to find the values of unknown variables that satisfy multiple equations at the same time. In the context of kinematic problems, one typically develops equations based on different events or time instances and solves them simultaneously to determine unknown parameters such as accelerations, times, or positions.

Relative Motion and Time Delay Analysis

Relative motion involves analyzing the movements of objects with respect to one another rather than independently. This concept is particularly useful when dealing with events such as vehicles passing predetermined points at different times. By considering the time delays between such events, one can set up relationships that describe how the positions of the moving objects relate, thereby enabling the calculation of quantities like acceleration and meeting time.

Kinematic Equations of Motion

The kinematic equations relate displacement, initial velocity, acceleration, and time for objects moving under constant acceleration. These equations serve as the fundamental tools for solving problems in one-dimensional motion by providing relationships that can be used to determine unknown quantities when certain variables are known.

Uniform Acceleration

Uniform acceleration is the condition in which an object's acceleration remains constant over time. This allows the use of simple kinematic equations to predict future positions and velocities based on the object's initial speed, acceleration, and elapsed time.

Unit Conversions

In problems involving motion, it is often necessary to convert units—for example, converting speeds from miles per hour to feet per second—to ensure consistency when applying the kinematic equations. Proper unit conversion is crucial to avoid errors and to make sure all terms in the equations are in compatible units.

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