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Chapter 12: Problem 39

Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. $$\mathbf{r}(t)=\left\langle t^{3}, t^{2}\right\rangle$$

Short Answer

Expert verified
Answer: The tangential component of acceleration, \(a_{T}(t)\), is given by \(\frac{12t^{3}+4t^{2}}{\sqrt{9t^{4} + 4t^{2}}}\), and the normal component of acceleration, \(a_{N}(t)\), is given by \(\frac{2}{\sqrt{9t^{4} + 4t^{2}}}\).

Step by step solution

01

Find the first and second derivatives of the position vector

Differentiate the position vector \(\mathbf{r}(t)\) with respect to time \(t\), to find the velocity vector, \(\mathbf{v}(t)\), and then differentiate \(\mathbf{v}(t)\) to find the acceleration vector, \(\mathbf{a}(t)\). $$\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} = \left\langle 3t^{2}, 2t\right\rangle$$ $$\mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt} = \left\langle 6t, 2\right\rangle$$
02

Calculate the magnitude of the velocity vector

Determine the magnitude of \(\mathbf{v}(t)\) using the formula: $$||\mathbf{v}(t)|| = \sqrt{(3t^{2})^{2} + (2t)^{2}} = \sqrt{9t^{4} + 4t^{2}}$$
03

Determine the unit tangent vector

Divide the velocity vector, \(\mathbf{v}(t)\), by its magnitude to find the unit tangent vector, \(\mathbf{T}(t)\). $$\mathbf{T}(t) = \frac{\mathbf{v}(t)}{||\mathbf{v}(t)||} = \frac{\left\langle 3t^{2}, 2t\right\rangle}{\sqrt{9t^{4} + 4t^{2}}}$$
04

Calculate the tangential component of the acceleration

Find the tangential component of the acceleration, \(a_{T}(t)\), using the dot product of the acceleration vector, \(\mathbf{a}(t)\), and the unit tangent vector, \(\mathbf{T}(t)\). $$a_{T}(t) = \mathbf{a}(t) \cdot \mathbf{T}(t) = \left\langle 6t, 2\right\rangle \cdot \frac{\left\langle 3t^{2}, 2t\right\rangle}{\sqrt{9t^{4} + 4t^{2}}} = \frac{12t^{3}+4t^{2}}{\sqrt{9t^{4} + 4t^{2}}}$$
05

Calculate the normal component of the acceleration

Use the formula \(a_{N}(t) = \sqrt{a(t)^{2} - a_{T}(t)^{2}}\) to find the normal component of the acceleration. $$a_{N}(t) = \sqrt{||\mathbf{a}(t)||^{2} - a_{T}(t)^{2}} = \sqrt{\left( (6t)^{2} + (2)^{2} \right) - \left( \frac{12t^{3}+4t^{2}}{\sqrt{9t^{4} + 4t^{2}}} \right)^{2}} = \frac{2}{\sqrt{9t^{4} + 4t^{2}}}$$
06

Present the tangential and normal components of acceleration

The tangential component of acceleration, \(a_{T}(t)\), is given by: $$a_{T}(t) = \frac{12t^{3}+4t^{2}}{\sqrt{9t^{4} + 4t^{2}}}$$ And the normal component of acceleration, \(a_{N}(t)\), is given by: $$a_{N}(t) = \frac{2}{\sqrt{9t^{4} + 4t^{2}}}$$

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Most popular questions from this chapter

Show that the two-dimensional trajectory $$x(t)=u_{0} t+x_{0}\( and \)y(t)=-\frac{g t^{2}}{2}+v_{0} t+y_{0},\( for \)0 \leq t \leq T$$ of an object moving in a gravitational field is a segment of a parabola for some value of \(T>0 .\) Find \(T\) such that \(y(T)=0\)

Evaluate the following limits. $$\lim _{t \rightarrow 0}\left(\frac{\sin t}{t} \mathbf{i}-\frac{e^{t}-t-1}{t} \mathbf{j}+\frac{\cos t+t^{2} / 2-1}{t^{2}} \mathbf{k}\right)$$

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3,0\rangle, \mathbf{v}=\langle 1,1,1\rangle\)

Zero curvature Prove that the curve $$ \mathbf{r}(t)=\left\langle a+b t^{p}, c+d t^{p}, e+f t^{p}\right\rangle $$ where \(a, b, c, d, e,\) and \(f\) are real numbers and \(p\) is a positive integer, has zero curvature. Give an explanation.

The points \(P, Q, R,\) and \(S,\) joined by the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{x},\) are the vertices of a quadrilateral in \(\mathrm{R}^{3}\). The four points needn't lie in \(a\) plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. a. Use vector addition to show that \(\mathbf{u}+\mathbf{v}=\mathbf{w}+\mathbf{x}\) b. Let \(m\) be the vector that joins the midpoints of \(P Q\) and \(Q R\) Show that \(\mathbf{m}=(\mathbf{u}+\mathbf{v}) / 2\) c. Let n be the vector that joins the midpoints of \(P S\) and \(S R\). Show that \(\mathbf{n}=(\mathbf{x}+\mathbf{w}) / 2\) d. Combine parts (a), (b), and (c) to conclude that \(\mathbf{m}=\mathbf{n}\) e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.
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