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Exercises for Section 13.5

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    Finding Components of Acceleration & Kepler's Laws

    1) Find the tangential and normal components of acceleration for \(\vecs r(t)=t^2\,\hat{\mathbf{i}}+2t \,\hat{\mathbf{j}}\) when \(t=1\).

    \(a_\vecs{T}=\sqrt{2}, \quad a_\vecs{N}=\sqrt{2}\)

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

    2) \(\vecs r(t)=⟨\cos(2t),\,\sin(2t),1⟩\)

    3) \(\vecs r(t)=⟨e^t \cos t,\,e^t\sin t,\,e^t⟩\). The graph is shown here:


    \(a_\vecs{T}=\sqrt{3}e^t, \quad a_\vecs{N}=\sqrt{2}e^t\)

    4) \(\vecs r(t)=⟨\frac{2}{3}(1+t)^{3/2}, \,\frac{2}{3}(1-t)^{3/2},\,\sqrt{2}t⟩\)

    5) \(\vecs r(t)=\left\langle 2t,\,t^2,\,\dfrac{t^3}{3}\right\rangle\)

    \(a_\vecs{T}=2t, \quad a_\vecs{N}=2\)

    6) \(\vecs r(t)=t^2\,\hat{\mathbf{i}}+t^2\,\hat{\mathbf{j}}+t^3\,\hat{\mathbf{k}}\)

    7) \(\vecs r(t)=⟨6t,\,3t^2,\,2t^3⟩\)

    \(a_\vecs{T}=\dfrac{6t +12t^3}{\sqrt{1+t^2+t^4}}, \quad a_\vecs{N}=6\sqrt{\dfrac{1+4t^2+t^4}{1+t^2+t^4}}\)

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

    \(a_\vecs{T}=0, \quad a_\vecs{N}=12\pi^2\)

    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 \(t=0\).

    \(a_\vecs{T}=0, \quad a_\vecs{N}=aω^2\)

    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 \(t=1.5\).

    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, \(\vecs F=m\vecs a\).)

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

    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.

    \(10.94\) 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.



    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

    Exercises for Section 13.5 is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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