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Mathematics LibreTexts

12.5E: 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) \vecs r(t)=3\cos(2πt)\,\hat{\mathbf{i}}+3\sin(2πt)\,\hat{\mathbf{j}}

Answer:
a_\vecs{T}=0, \quad a_\vecs{N}=12\pi^2

9) Find the tangential and normal components of acceleration for \vecs r(t)=a\cos(ωt)\,\hat{\mathbf{i}}+b\sin(ωt)\,\hat{\mathbf{j}} at t=0.

Answer:
a_\vecs{T}=0, \quad a_\vecs{N}=aω^2

10) Suppose that the position function for an object in three dimensions is given by the equation \vecs r(t)=t\cos(t)\,\hat{\mathbf{i}}+t\sin(t)\,\hat{\mathbf{j}}+3t\,\hat{\mathbf{k}}.

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. a_\vecs{T}=0.43\,\text{m/sec}^2, \quad a_\vecs{N}=2.46\,\text{m/sec}^2

11) The force on a particle is given by \vecs f(t)=(\cos t)\,\hat{\mathbf{i}}+(\sin t)\,\hat{\mathbf{j}}. The particle is located at point (c,0) at t=0. The initial velocity of the particle is given by \vecs v(0)=v_0\,\hat{\mathbf{j}}. Find the path of the particle of mass m. (Recall, \vecs F=m\vecs a.)

Answer:
\vecs r(t)=\left(\dfrac{-\cos t}{m}+c+\frac{1}{m}\right)\,\hat{\mathbf{i}}+\left(\dfrac{−\sin t}{m}+\left(v_0+\frac{1}{m}\right)t\right)\,\hat{\mathbf{j}}

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 v_0=\sqrt{\dfrac{2GM}{r_0}} is the minimum speed needed when \theta=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 \theta=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.


This page titled 12.5E: Exercises for Section 12.5 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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