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Mathematics LibreTexts

1.5.5E: Exercises for Section 12.5

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Finding Components of Acceleration & Kepler's Laws

1) Find the tangential and normal components of acceleration for r(t)=t2ˆi+2tˆj when t=1.

Answer:
aT=2,aN=2

In questions 2 - 8, find the tangential and normal components of acceleration.

2) r(t)=cos(2t),sin(2t),1

3) r(t)=etcost,etsint,et. The graph is shown here:

CNX_Calc_Figure_13_04_210.jpg

Answer:
aT=3et,aN=2et

4) r(t)=23(1+t)3/2,23(1t)3/2,2t

5) r(t)=2t,t2,t33

Answer:
aT=2t,aN=2

6) r(t)=t2ˆi+t2ˆj+t3ˆk

7) r(t)=6t,3t2,2t3

Answer:
aT=6t+12t31+t2+t4,aN=61+4t2+t41+t2+t4

8) r(t)=3cos(2πt)ˆi+3sin(2πt)ˆj

Answer:
aT=0,aN=12π2

9) Find the tangential and normal components of acceleration for r(t)=acos(ωt)ˆi+bsin(ωt)ˆj at t=0.

Answer:
aT=0,aN=aω2

10) Suppose that the position function for an object in three dimensions is given by the equation r(t)=tcos(t)ˆi+tsin(t)ˆj+3tˆk.

a. Show that the particle moves on a circular cone.

b. Find the angle between the velocity and acceleration vectors when t=1.5.

c. Find the tangential and normal components of acceleration when t=1.5.

Answer:
c. aT=0.43m/sec2,aN=2.46m/sec2

11) The force on a particle is given by f(t)=(cost)ˆi+(sint)ˆj. The particle is located at point (c,0) at t=0. The initial velocity of the particle is given by v(0)=v0ˆj. Find the path of the particle of mass m. (Recall, F=ma.)

Answer:
r(t)=(costm+c+1m)ˆi+(sintm+(v0+1m)t)ˆj

12) An automobile that weighs 2700 lb makes a turn on a flat road while traveling at 56 ft/sec. If the radius of the turn is 70 ft, what is the required frictional force to keep the car from skidding?

13) Using Kepler’s laws, it can be shown that v0=2GMr0 is the minimum speed needed when θ=0 so that an object will escape from the pull of a central force resulting from mass M. Use this result to find the minimum speed when θ=0 for a space capsule to escape from the gravitational pull of Earth if the probe is at an altitude of 300 km above Earth’s surface.

Answer:
10.94 km/sec

14) Find the time in years it takes the dwarf planet Pluto to make one orbit about the Sun given that a=39.5 A.U.

 

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


1.5.5E: Exercises for Section 12.5 is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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