Collision courses. Initially two particles are at positions x1=5 cm, y1=0 ; and x2=0, y2=10 cm

step 1 In this question we have given two blocks and we have to find the minimum value of this force F

Collision courses. Initially two particles are at positions...

Collision courses. Initially two particles are at positions $x_{1}=5 \mathrm{~cm}, y_{1}=0 ;$ and $x_{2}=0, y_{2}=10 \mathrm{~cm}$, with $\mathrm{v}_{1}=-4 \times$ $10^{4} \hat{\mathbf{x}} \mathrm{cm} / \mathrm{s}$, and $\mathbf{v}_{2}$ is along $-\hat{\mathrm{y}}$ as in Fig. $3.27$. (a) What must be the value of $v_{2}$ if they are to collide? Ans. $-8 \times 10^{4} \hat{\mathbf{y}} \mathrm{cm} / \mathrm{s}$ (b) What is the value of $v_{r}$, the relative velocity? Ans. $4 \times 10^{4}(2 \hat{\mathbf{y}}-\hat{\mathbf{x}}) \mathrm{cm} / \mathrm{s}$ (c) Establish a general criterion for recognizing a collision course for two objects in terms of their positions $\mathbf{r}_{1}, \mathbf{r}_{2}$ and velocities $\mathbf{v}_{1}, \mathbf{v}_{2^{+}}$

Collision courses. Initially two particles are at positions $x_{1}=5 \mathrm{~cm}, y_{1}=0 ;$ and $x_{2}=0, y_{2}=10 \mathrm{~cm}$, with $\mathrm{v}_{1}=-4 \times$
$10^{4} \hat{\mathbf{x}} \mathrm{cm} / \mathrm{s}$, and $\mathbf{v}_{2}$ is along $-\hat{\mathrm{y}}$ as in Fig. $3.27$.
(a) What must be the value of $v_{2}$ if they are to collide? Ans. $-8 \times 10^{4} \hat{\mathbf{y}} \mathrm{cm} / \mathrm{s}$
(b) What is the value of $v_{r}$, the relative velocity? Ans. $4 \times 10^{4}(2 \hat{\mathbf{y}}-\hat{\mathbf{x}}) \mathrm{cm} / \mathrm{s}$
(c) Establish a general criterion for recognizing a collision course for two objects in terms of their positions $\mathbf{r}_{1}, \mathbf{r}_{2}$ and velocities $\mathbf{v}_{1}, \mathbf{v}_{2^{+}}$

Key Concepts

Dot Product Criterion for Collision

A key insight in determining a collision is using the dot product between the relative position vector and the relative velocity vector. If this dot product is negative, it indicates that the objects are moving towards each other, thus satisfying a necessary condition for a potential collision. This criterion provides a simple mathematical check to see if the objects are on a collision course.

Relative Motion Analysis

This concept involves studying the motion of one object with respect to another by looking at their relative position and velocity vectors. Instead of analyzing each object's motion in isolation, one subtracts one object’s motion from the other’s, leading to a simplified problem where one object is stationary and the other moves with the difference in velocities. This approach is particularly useful in determining if and when two moving objects might meet.

Parametric Equations of Motion

In kinematics, the positions of moving objects are frequently described by parametric equations, where the position vector is expressed as a function of time by adding the product of velocity and time to the initial position. This formulation enables one to set up equations for each dimension and solve for the time and position at which a collision might occur.

Vector Components and Two-Dimensional Motion

Analyzing motion in a plane requires breaking down position and velocity into their component forms along mutually perpendicular axes (typically x and y). Understanding how components behave independently allows for the solution of problems involving collisions or intersections by analyzing each direction separately.

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