Constant Acceleration Cars A and B move in the same...

Constant Acceleration Cars $A$ and $B$ move in the same direction in adjacent lanes.The position $x$ of car $A$ is given in Fig. $2-30,$ from time $t=0$ to $t=7.0 \mathrm{s}$ The figure's vertical scaling is set by $x_{s}=$ 32.0 $\mathrm{m} .$ At $t=0,$ car $B$ is at $x=$ 0 , with a velocity of 12 $\mathrm{m} / \mathrm{s}$ and a negative constant acceleration $a_{B} .$ (a) What must $a_{B}$ be such that the cars are (momentarily side by side (momentarily at the same value of $x )$ at $t=4.0 \mathrm{s}$ ? (b) For that value of $a_{B}$ , how many times are the cars side by side? (c) Sketch the position $x$ of car $B$ versus time $t$ on Fig. $2-30 .$ How many times will the cars be side by side if the magnitude of acceleration $a_{B}$ is $(\mathrm{d})$ more than and $(\mathrm{e})$ less than the answer to part $(\mathrm{a}) ?$

Constant Acceleration
Cars $A$ and $B$ move in the same direction in adjacent lanes.The position $x$ of car $A$ is given in Fig. $2-30,$ from time $t=0$ to $t=7.0 \mathrm{s}$ The figure's vertical scaling is set by $x_{s}=$
32.0 $\mathrm{m} .$ At $t=0,$ car $B$ is at $x=$ 0 , with a velocity of 12 $\mathrm{m} / \mathrm{s}$ and a negative constant acceleration $a_{B} .$ (a) What must $a_{B}$ be such that the cars are (momentarily side by side (momentarily at the same value of $x )$ at $t=4.0 \mathrm{s}$ ? (b) For that value of $a_{B}$ , how many times are the cars side by side? (c) Sketch the position $x$ of car $B$ versus time $t$ on Fig. $2-30 .$ How many times will the cars be side by side if the magnitude of acceleration $a_{B}$ is $(\mathrm{d})$ more than and $(\mathrm{e})$ less than the answer to part $(\mathrm{a}) ?$

Key Concepts

Constant Acceleration Kinematics

This concept deals with the motion of objects under a constant acceleration. The standard kinematic equations relate displacement, initial velocity, time, acceleration, and final velocity. These equations are fundamental when analyzing situations where an object’s acceleration does not change over time, allowing for predictions of position and velocity at any given moment.

Position–Time Equation

The position–time equation, typically written as x = x? + v?t + (1/2)at², is central to solving problems involving constant acceleration. It provides a direct relationship between an object's initial conditions (position and velocity) and its acceleration, enabling the determination of how far it has traveled after a given time.

Relative Motion Analysis

Relative motion analysis involves comparing the positions of two or more objects moving in a common frame of reference. In problems where two objects are moving, determining when they are 'side by side' requires setting their position equations equal to each other, thus analyzing their motion relative to one another.

Quadratic Equations in Kinematics

When equating the position equations of two objects in relative motion, the resulting expression is often a quadratic equation in time. The solution of this quadratic equation—especially the discriminant—provides valuable insights into the number and type of intersections (i.e., times when the objects have the same position), which is crucial for understanding collision or overtaking scenarios.

Graphical Analysis of Motion

Graphical analysis involves plotting position versus time for moving objects to visually interpret aspects of their motion. By examining the curves, one can identify intersections and critical points, such as when two objects are side by side. This technique serves as a valuable tool for verifying algebraic solutions and gaining intuitive understanding of the behavior of moving objects.

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