Chapter 4, Problem 81GP

Collision Course A useful rule of thumb in piloting is that if the heading from your airplane to a second airplane remains constant, the two airplanes are on a collision course. Consider the two airplanes shown in Figure 4-33. At time t = 0 airplane 1 is at the location (X, 0) and moving in the positive y direction; airplane 2 is at (0, Y) and moving in the positive x direction. The speed of airplane 1 is v₁. (a) What speed must airplane 2 have if the airplanes are to collide at the point (X, Y)? (b) Assuming airplane 2 has the speed found in part (a), calculate the displacement from airplane 1 to airplane 2, $\Delta {\stackrel{\to }{\text{r}}\text{ = }\stackrel{\to }{\text{r}}}_{\text{2}}-{\stackrel{\to }{\text{r}}}_1$. (c) Use your results from part (b) to show that ${\left(\Delta r\right)}_y/{\left(\Delta r\right)}_x=-Y/X$, independent of time. This shows that $\Delta {\stackrel{\to }{\text{r}}\text{ = }\stackrel{\to }{\text{r}}}_{\text{2}}-{\stackrel{\to }{\text{r}}}_1$ maintains a constant direction until the collision, as specified in the rule of thumb.

Figure 4-33

Problem 81