Finding the point in time where two objects collide

Question

Suppose you have two boats $A$ and $B$ both travelling at a constant velocity; let's say $A$ travels at a velocity of $(-\mathbf{i}+6\mathbf{j})$ m/s, and $B$ at $(3\mathbf{i}+4\mathbf{j})$ m/s, and when $t=0$, boat $A$ has position vector of $(2\mathbf{i}-10\mathbf{j})$ m, and boat $B$ has a position vector of $(-26\mathbf{i}+4\mathbf{j})$ m. Is there a method to find the time at which these boats will collide?

Answer 1

$$x_A=2-t $$ $$y_A=-10+6t $$

$$x_B=-26+3t $$ $$y_B=4+4t $$

They collide when $x_A=x_B $

or $$2-t=-26+3t \implies \color {red}{t=7 s}$$

if we plugg it into $y_A$, we get

$$y_A=-10+42=32=y_B $$They will meet each other at the point $$(-5 ,32)$$

Answer 2

Guide:

$$\vec{s}_t=\vec{s}_0+t\vec{v}$$

You should be able to find locations of both boats at any moment $t$ and solve for collision time.

Answer 3

HINT

the motion of $A$ is given by: $$ \begin{pmatrix} x_A\\y_A \end{pmatrix}= \begin{pmatrix} -1\\6 \end{pmatrix}t+\begin{pmatrix} 2\\-10 \end{pmatrix} $$ and the motion of $B$ is $$ \begin{pmatrix} x_B\\y_B \end{pmatrix}= \begin{pmatrix} 3\\4 \end{pmatrix}t+\begin{pmatrix} -26\\4 \end{pmatrix} $$

and they collide if there is a time $t$ such that: $$ \begin{pmatrix} x_A\\y_A \end{pmatrix}=\begin{pmatrix} x_B\\y_B \end{pmatrix} $$

Answer 4

Since the time to collision is the same or constant, after finding length from orgin by radial distances $L_1,L_2 $ of position vector length

we should have

$$ t = \frac{L_1}{v_1}=\frac{L_2}{v_2} $$