T1W2L3: Relative Velocity (2)

Relative Velocity practice questions

• BvA is the velocity B relative to A

example questions §

• to a person on a ship travelling with velocity (15i + 2j) km/h the wind appears to have velocity (-9i + 2j) km/h. Find the true velocity of the wind
  • ${\vec{v}}_s=15\vec{i}+2\vec{j}$ km/h
  • ${\vec{v}}_w=?$
  • ${}_w{\vec{v}}_s={\vec{v}}_w-{\vec{v}}_s$
    • ${\vec{v}}_w{=}_w{\vec{v}}_s+{\vec{v}}_s$
    • $=-9\vec{i}+2\vec{j}+15\vec{i}+2\vec{j}$
    • $=6\vec{i}+4\vec{j}$ km/h
• the velocity of ship A, as seen by an observer on ship B, is 7i - 10j. The velocity of ship A, as seen by an observer on ship C, is 13i - 2j. Find the velocity on ship B as seen by an observer on C.
  • ${}_A{\vec{v}}_B=7i-10j$
  • ${}_A{\vec{v}}_C=13i-2j$
  • ${}_B{\vec{v}}_C=?$
  • ${}_A{\vec{v}}_B={\vec{v}}_A-{\vec{v}}_B$
  • ${}_A{\vec{v}}_C={\vec{v}}_A-{\vec{v}}_C$
    • ${\vec{v}}_A{=}_A{\vec{v}}_B+{\vec{v}}_B$
  • ${}_B{\vec{v}}_C={\vec{v}}_B-{\vec{v}}_C={\vec{v}}_A-{\vec{v}}_C-({\vec{v}}_A-{\vec{v}}_B)=13i-2j-(7i-10j)=6i+8j$ km/h
• harder exam question that goes beyond (conceptually) questions in sadler
  • at 10am, electronic hot wheels A and B are launched from their locations at (7i + 3j)m and (-i - 0.8j)m with constant velocities (-i+0.25j)m/s and (i+1.2j)m/s respectively.
    • write vector expressions for $r_a(t)$, the position vector of A at t seconds, and $r_b(t)$, the position vector of B at t seconds.
      • $r_a(t)=(7i+3j)+t(-i+0.25j)m=(7-t)i+(3+0.25t)j$
      • $r_b(t)=(-i+0.8j)+t(i+1.2j)m=(t-1)i+(-0.8+1.2t)j$
    • write a vector expression for ${}_A{\vec{r}}_B(t)$, the relative position of A with respect to B at t seconds.
      • ${}_A{\vec{r}}_B={\vec{r}}_A-{\vec{r}}_B$
      • ${}_A{\vec{r}}_B(t)=(7-t)i+0.25(3+t)j-((t-1)i+(-0.8+1.2t)j)$
      • therefore ${}_A{\vec{r}}_B(t)=(8-2t)i+(3.8-0.95t)j$
    • hence, show that the cards will collide if the se velocities are maintained. state when and where the collisions occur.
      • occurs at $t=4$, at $(3i,4j)$