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 $\vecs r(t)=t^2\,\hat{\mathbf{i}}+2t \,\hat{\mathbf{j}}$ when $t=1$.

Answer:: $a_\vecs{T}=\sqrt{2}, \quad a_\vecs{N}=\sqrt{2}$

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:

$CNX_Calc_Figure_13_04_210.jpg$

Answer:: $a_\vecs{T}=\sqrt{3}e^t, \quad a_\vecs{N}=\sqrt{2}e^t$

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

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

Answer:: $a_\vecs{T}=2t, \quad a_\vecs{N}=2$

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

7) $\vecs r(t)=⟨6t,\,3t^2,\,2t^3⟩$

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

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

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.