13.4 Motion Along A Space Curve: Velocity and Acceleration

1-5. Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for specified value of .


1. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-1.html

 Sol)

  

var('t')

r(t)=(t^3+1,t)

v=diff(r(t),t)

a=diff(v,t)

s=v.norm()

print v,a,s

v2=v.subs(t=2)

a2=a.subs(t=2)

print v2, a2

  (3*t^2, 1) (6*t, 0) sqrt(abs(3*t^2)^2 + 1)

  (12, 1) (12, 0)


  

p1=parametric_plot(r(t),(t,0,3))

p2=line([r(2),r(2)+v2], color='red')

p3=line([r(2),r(2)+a2], color='green')

show(p1+p2+p3)


 


2. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-2.html

Sol) 

  

var('t')

r(t)=(2+2*t,4*sqrt(t))

v=diff(r(t),t)

a=diff(v,t)

v1=v.subs(t=1)

a1=a.subs(t=1)

s=v.norm()

print v,a,s

  (2, 2/sqrt(t)) (0, -1/t^(3/2)) sqrt(abs(2/sqrt(t))^2 + 4)


  

p1=parametric_plot(r(t),(t,0,3))

p2=line([r(1),r(1)+v1], color='red')

p3=line([r(1),r(1)+a1], color='green')

show(p1+p2+p3)


 


3. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-3.html

 Sol)

  

var('t')

r(t)=(exp(2*t),exp(-t))

v=diff(r(t),t)

a=diff(v,t)

v0=v.subs(t=0)

a0=a.subs(t=0)

s=v.norm()

print v,a,s

  (2*e^(2*t), -e^(-t)) (4*e^(2*t), e^(-t)) sqrt(abs(-e^(-t))^2 + abs(2*e^(2*t))^2)


  

p1=parametric_plot(r(t),(t,-1,1))

p2=line([r(0),r(0)+v0], color='red')

p3=line([r(0),r(0)+a0], color='green')

show(p1+p2+p3)


 


4. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-4.html

 Sol)

  

var('t')

r(t)=(2*sin(t),cos(t))

v=diff(r(t),t)

a=diff(v,t)

vt=v.subs(t=pi/3)

at=a.subs(t=pi/3)

s=v.norm()

print v,a,s

  (2*cos(t), -sin(t)) (-2*sin(t), -cos(t)) sqrt(abs(-sin(t))^2 + abs(2*cos(t))^2)


  

p1=parametric_plot(r(t),(t,0,3))

p2=line([r(pi/3),r(pi/3)+vt], color='red')

p3=line([r(pi/3),r(pi/3)+at], color='green')

show(p1+p2+p3)


 


5. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-5.html

 Sol)

  

var('t')

r(t)=(3*sin(t),t,3*cos(t))

v=diff(r(t),t)

a=diff(v,t)

v0=v.subs(t=0)

a0=a.subs(t=0)

s=v.norm()

print v,a,s

  (3*cos(t), 1, -3*sin(t)) (-3*sin(t), 0, -3*cos(t)) sqrt(abs(-3*sin(t))^2 + abs(3*cos(t))^2 + 1)


  

p1=parametric_plot3d(r(t),(t,-1,2))

p2=line([r(0),r(0)+v0], color='red')

p3=line([r(0),r(0)+a0], color='green')

show(p1+p2+p3)

 


6-10. Find the velocity, acceleration and speed of a particle with the given position function.


6. .

 Sol) ,

      ,

      .


7. .

 Sol) ,

      ,

      .


8. .

 Sol) ,

      ,

      .


9. .

 Sol) ,

      ,

      .


10. .

 Sol) ,

      ,

      .


11-12. Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.


11. .

 Sol) . Since , we have .

      . Since , we have .


12. .

 Sol) . Since , we have .

      .

      Since , we have .



13-14. Find the position vector of a particle that has the given acceleration and the specified initial velocity and position.


13. .

 Sol) . Since ,

      we have .

      .

      Since , we have .


14. .

 Sol) .

      Since , we have .

      .

      Since , we have .


15. The position function of a particle is given by . When is the speed a minimum?

http://matrix.skku.ac.kr/cal-lab/cal-13-4-15.html

 Sol)

  

var('t')

r(t)=(t^2,t,t^2-4*t)

v=diff(r(t),t)

s=v.norm()

s0=diff(s,t)

solve(s0==0,t)

  t=1


  

plot(s,(t,0,3))


 


16-20. Find the tangential and normal components of the acceleration vector.


16. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-16.html

 Sol)

 

var('t')

r(t)=(t-t^3,t^2,0)

dr=diff(r(t),t)

ddr=diff(r(t),t,2)

T=dr.dot_product(ddr)/dr.norm()

print T

N=(dr.cross_product(ddr)).norm() / dr.norm()

print N

  2*(3*(3*t^2 - 1)*t + 2*t)/sqrt(abs(-3*t^2 + 1)^2 + abs(2*t)^2)

  sqrt(abs(6*t^2 + 2)^2)/sqrt(abs(-3*t^2 + 1)^2 + abs(2*t)^2)


17. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-17.html

 Sol)

  

var('t')

r(t)=(cos(t),sin(t),2*t)

dr=diff(r(t),t)

ddr=diff(r(t),t,2)

T=dr.dot_product(ddr)/dr.norm()

print T

N=(dr.cross_product(ddr)).norm() / dr.norm()

print N

  0

  sqrt(abs(sin(t)^2 + cos(t)^2)^2 + abs(2*sin(t))^2 + abs(-2*cos(t))^2)/sqrt(abs(-sin(t))^2 + abs(cos(t))^2 + 4)


18. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-18.html


 Sol)

  

var('t')

r(t)=(2*t,t^2,t^3)

dr=diff(r(t),t)

ddr=diff(r(t),t,2)

T=dr.dot_product(ddr)/dr.norm()

print T

N=(dr.cross_product(ddr)).norm() / dr.norm()

print N

  2*(9*t^3 + 2*t)/sqrt(abs(3*t^2)^2 + abs(2*t)^2 + 4)

  sqrt(abs(6*t^2)^2 + abs(-12*t)^2 + 16)/sqrt(abs(3*t^2)^2 + abs(2*t)^2 + 4)

 

19. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-19.html

 Sol)

  

var('t')

r(t)=(exp(2*t),t,exp(-2*t))

dr=diff(r(t),t)

ddr=diff(r(t),t,2)

T=dr.dot_product(ddr)/dr.norm()

print T

N=(dr.cross_product(ddr)).norm() / dr.norm()

print N

  -8*(e^(-4*t) - e^(4*t))/sqrt(abs(-2*e^(-2*t))^2 + abs(2*e^(2*t))^2 + 1)

  sqrt(abs(4*e^(-2*t))^2 + abs(-4*e^(2*t))^2 + 256)/sqrt(abs(-2*e^(-2*t))^2 + abs(2*e^(2*t))^2 + 1)

 

20. .

http://matrix.skku.ac.kr/cal-lab/cal-13-4-20.html

 Sol)

  

var('t')

r(t)=(t,1/2*(cos(t))^2,1/2*(sin(t))^2)

dr=diff(r(t),t)

ddr=diff(r(t),t,2)

T=dr.dot_product(ddr)/dr.norm()

print T

N=(dr.cross_product(ddr)).norm() / dr.norm()

print N

  -2*(sin(t)^2 - cos(t)^2)*sin(t)*cos(t)/sqrt(abs(-sin(t)*cos(t))^2 + abs(sin(t)*cos(t))^2 + 1)

  sqrt(2)*sqrt(abs(sin(t)^2 - cos(t)^2)^2)/sqrt(abs(-sin(t)*cos(t))^2 + abs(sin(t)*cos(t))^2 + 1)