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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) $$\vecs r(t)=⟨2t,t^2,\frac{t^3}{3}⟩$$

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

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

7) $$\vecs r(t)=⟨6t,3t^2,2t^3⟩$$

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

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

$$a_\vecs{T}=0, \quad a_\vecs{N}=2\sqrt{3}\pi$$

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 $$\vecs f(t)=(cost)\,\hat{\mathbf{i}}+(sint)\,\hat{\mathbf{j}}$$. 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, $$\vecs F=m\vecs a$$.)

$$\vecs r(t)=\left(\frac{-1}{m}\cos t+c+\frac{1}{m}\right)\,\hat{\mathbf{i}}+\left(\frac{−\sin t}{m}+\left(v_0+\frac{1}{m}\right)t\right)\,\hat{\mathbf{j}}$$
13) Using Kepler’s laws, it can be shown that $$v_0=\sqrt{\frac{2GM}{r_0}}$$ 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.