Two particles have position vectors given by:
[tex]\begin{array}{l}
\overrightarrow{r_1}=2 t i-t^2 j+\left(3 t^2-4 t\right) k \\
\overrightarrow{r_2}=(5 t-12 t+4) i+t^3 j-3 t k
\end{array}[/tex]
Express in vector form, the relative velocity of the 2nd particle with respect to the 1st particle.
Answer (2)
To express the relative velocity of the second particle with respect to the first, we calculated the velocity vectors by differentiating the position vectors of both particles. The resulting relative velocity is given by v 21 = − 9 i + ( 3 t 2 + 2 t ) j + ( 1 − 6 t ) k . This signifies the change in velocity of the second particle as observed from the first particle's frame of reference.
;
Find the velocity vector of particle 1 by differentiating its position vector with respect to time: v 1 = 2 i − 2 t j + ( 6 t − 4 ) k .
Find the velocity vector of particle 2 by differentiating its position vector with respect to time: v 2 = − 7 i + 3 t 2 j − 3 k .
Calculate the relative velocity of particle 2 with respect to particle 1: v 21 = v 2 − v 1 .
Express the relative velocity in vector form: v 21 = − 9 i + ( 3 t 2 + 2 t ) j + ( 1 − 6 t ) k . The relative velocity of the 2nd particle with respect to the 1st particle is − 9 i + ( 3 t 2 + 2 t ) j + ( 1 − 6 t ) k .
Explanation
Problem Analysis We are given the position vectors of two particles and asked to find the relative velocity of the second particle with respect to the first. This involves finding the velocity vectors of each particle by differentiating their position vectors with respect to time, and then subtracting the velocity vector of the first particle from that of the second particle.
Velocity of Particle 1 The position vector of the first particle is given by r 1 = 2 t i − t 2 j + ( 3 t 2 − 4 t ) k . To find the velocity vector v 1 , we differentiate r 1 with respect to time t :
v 1 = d t d r 1 = d t d ( 2 t i − t 2 j + ( 3 t 2 − 4 t ) k ) = 2 i − 2 t j + ( 6 t − 4 ) k
Velocity of Particle 2 The position vector of the second particle is given by r 2 = ( 5 t − 12 t + 4 ) i + t 3 j − 3 t k = ( − 7 t + 4 ) i + t 3 j − 3 t k . To find the velocity vector v 2 , we differentiate r 2 with respect to time t :
v 2 = d t d r 2 = d t d (( − 7 t + 4 ) i + t 3 j − 3 t k ) = − 7 i + 3 t 2 j − 3 k
Relative Velocity The relative velocity of the second particle with respect to the first particle is given by v 21 = v 2 − v 1 . Therefore, v 21 = ( − 7 i + 3 t 2 j − 3 k ) − ( 2 i − 2 t j + ( 6 t − 4 ) k ) = ( − 7 − 2 ) i + ( 3 t 2 − ( − 2 t )) j + ( − 3 − ( 6 t − 4 )) k = − 9 i + ( 3 t 2 + 2 t ) j + ( 1 − 6 t ) k
Final Answer The relative velocity of the 2nd particle with respect to the 1st particle is v 21 = − 9 i + ( 3 t 2 + 2 t ) j + ( 1 − 6 t ) k .
Examples
Understanding relative motion is crucial in many real-world applications, such as air traffic control. Air traffic controllers must calculate the relative velocities of aircraft to ensure safe separation and prevent collisions. Similarly, in robotics, understanding relative velocities is essential for coordinating the movements of multiple robots in a shared workspace, allowing them to perform tasks efficiently without interfering with each other.
