d) for ( 0 leq t leq 10 ), particles ( p ) and ( q ) move along the ( x ) axis. the position of particle ( p…

d) for ( 0 leq t leq 10 ), particles ( p ) and ( q ) move along the ( x ) axis. the position of particle ( p ) can be modeled by ( x_{p}(t) ) as shown in the figure above. the position of particle ( q ) is defined by ( x_{q}(t) ). selected values of ( x_{q}(t), v_{q}(t) ), and ( a_{q}(t) ) are given in the table above. at what time ( t ) are particles ( p ) and ( q ) moving towards each other?

d) for ( 0 leq t leq 10 ), particles ( p ) and ( q ) move along the ( x ) axis. the position of particle ( p ) can be modeled by ( x_{p}(t) ) as shown in the figure above. the position of particle ( q ) is defined by ( x_{q}(t) ). selected values of ( x_{q}(t), v_{q}(t) ), and ( a_{q}(t) ) are given in the table above. at what time ( t ) are particles ( p ) and ( q ) moving towards each other?

Answer

Explanation:

Step1: Find the velocity of particle (P)

The velocity of particle (P), (v_{P}(t)) is the slope of the position - time graph (x_{P}(t)). For (0\leq t\leq3), (v_{P}(t)=\frac{0 - 3}{3-0}=- 1) For (3\leq t\leq8), (v_{P}(t)=\frac{0.5+1}{8 - 3}=\frac{1.5}{5}=0.3) For (8\leq t\leq10), (v_{P}(t)=\frac{-3 - 0.5}{10 - 8}=\frac{-3.5}{2}=-1.75)

Step2: Analyze the condition for moving towards each other

Two particles are moving towards each other when (v_{P}(t)\times v_{Q}(t)<0) (opposite - direction velocities)

  • At (t = 1): (v_{P}(1)=-1), (v_{Q}(1) = 3), and (v_{P}(1)\times v_{Q}(1)=(-1)\times3=-3<0)
  • At (t = 3): (v_{P}(3)=-1), (v_{Q}(3)=0), and (v_{P}(3)\times v_{Q}(3)=(-1)\times0 = 0)
  • At (t = 7): (v_{P}(7)=0.3), (v_{Q}(7)=-2), and (v_{P}(7)\times v_{Q}(7)=(0.3)\times(-2)=-0.6<0)
  • At (t = 9): (v_{P}(9)=-1.75), (v_{Q}(9)=3), and (v_{P}(9)\times v_{Q}(9)=(-1.75)\times3=-5.25<0)

Answer:

(t = 1), (t = 7), (t = 9)