# Assume that an object moves on a trajectory according to the equations x(t)=t+cost, y(t)=t−sint. Find the magnitude of the acceleration vector. Question
Equations Assume that an object moves on a trajectory according to the equations $$\displaystyle{x}{\left({t}\right)}={t}+{\cos{{t}}},{y}{\left({t}\right)}={t}−{\sin{{t}}}$$. Find the magnitude of the acceleration vector. 2020-11-13
Step 1
An object moves on a trajectory according to the equations
$$\displaystyle{x}{\left({t}\right)}={t}+{\cos{{t}}},{y}{\left({t}\right)}={t}-{\sin{{t}}}$$
Differentiating with respect to t, we get
$$\displaystyle{x}'{\left({t}\right)}={1}-{\sin{{t}}},{y}'{\left({t}\right)}={1}-{\cos{{t}}}$$
Again differentiating with respect to t, we get
$$\displaystyle{x}{''}{\left({t}\right)}=-{\cos{{t}}},{y}{''}{\left({t}\right)}={\sin{{t}}}$$
Step 2
Now, acceleration vector is
$$\displaystyle\vec{{{a}}}{\left({t}\right)}={x}{''}{\left({t}\right)}\hat{{{i}}}+{y}{''}{\left({t}\right)}\hat{{{j}}}$$
$$\displaystyle\vec{{{a}}}{\left({t}\right)}=-\hat{{{i}}}{\cos{{t}}}+\hat{{{j}}}{\sin{{t}}}$$
Magnitude of acceleration vector is evaluated as follows.
$$\displaystyle{\left|{\vec{{{a}}}{\left({t}\right)}}\right|}=\sqrt{{{\left(-{\cos{{t}}}\right)}^{{2}}+{\left({\sin{{t}}}\right)}^{{2}}}}=\sqrt{{{{\cos}^{{2}}{t}}+{{\sin}^{{2}}{t}}}}={1}$$
Step 3
Result: Magnitude of acceleration vector is 1 unit.

### Relevant Questions 1)What is the position vector r(t) as a function of angle $$\displaystyle\theta{\left({t}\right)}$$. For later remember that $$\displaystyle\theta{\left({t}\right)}$$ is itself a function of time.
Give your answer in terms of $$\displaystyle{R},\theta{\left({t}\right)}$$, and unit vectors x and y corresponding to the coordinate system in thefigure. 2)For uniform circular motion, find $$\displaystyle\theta{\left({t}\right)}$$ at an arbitrary time t.
Give your answer in terms of $$\displaystyle\omega$$ and t.
3)Find r, a position vector at time.
Give your answer in terms of R and unit vectors x and/or y.
4)Determine an expression for the positionvector of a particle that starts on the positive y axis at (i.e., at ,(x_{0},y_{0})=(0,R)) and subsequently moves with constant $$\displaystyle\omega$$.
Express your answer in terms of R, \omega ,t ,and unit vectors x and Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
$$\displaystyle{x}=-{7}{\cos{{2}}}{t},{y}=-{7}{\sin{{2}}}{t},{0}\le{t}\le\pi$$ a) Find parametric equations for the line. (Use the parameter t.)
The line of intersection of the planes
x + y + z = 2 and x + z = 0
(x(t), y(t), z(t)) =
b) Find the symmetric equations. Consider the following system of llinear equations.
$$\displaystyle\frac{{1}}{{3}}{x}+{y}=\frac{{5}}{{4}}$$
$$\displaystyle\frac{{2}}{{3}}{x}-\frac{{4}}{{3}}{y}=\frac{{5}}{{3}}$$
Part A: $$\displaystyle\frac{{{W}\hat{\propto}{e}{r}{t}{y}}}{{\propto{e}{r}{t}{i}{e}{s}}}$$ can be used to write an equivalent system?
Part B: Write an equivalent system and use elimination method to solve for x and y. -In the system of equations in x and y, 2x+3y=12, 4x+ay=16, where a is an integer, what would a have to be for the equations to be inconsistent?
-In the system of equations in x and y, 2x+3y=12, 4x+ay=b, where a is the answer to question 2, what would b be if the equations are dependent? decide if the given statement is true or false,and give a brief justiﬁcation for your answer.If true, you can quote a relevant deﬁnition or theorem. If false,provide an example,illustration,or brief explanation of why the statement is false.
Q) The system of differential equations $$\displaystyle{x}′={t}{y}+{5}{\left({\sin{{t}}}\right)}{y}−{3}{e}^{{t}},{y}′={t}^{{2}}{x}−{7}{x}{y}−{1}$$ is a ﬁrst-order linear system of differential equations. 1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?” Use a triple integral to find the volume of the solid bounded by the graphs of the equations. $$\displaystyle{z}={2}−{y},{z}={4}−{y}^{{2}},{x}={0},{x}={3},{y}={0}$$ The position vector $$\displaystyle{r}{\left({t}\right)}={\left\langle{n}{t},\frac{{1}}{{t}^{{2}}},{t}^{{4}}\right\rangle}$$ describes the path of an object moving in space.
(b) Evaluate the velocity vector and acceleration vector of the object at the given value of $$\displaystyle{t}=\sqrt{{3}}$$ The position vector $$\displaystyle{r}{\left({t}\right)}={\left\langle{\ln{{t}}},\frac{{1}}{{t}^{{2}}},{t}^{{4}}\right\rangle}$$ describes the path of an object moving in space.
(b) Evaluate the velocity vector and acceleration vector of the object at the given value of $$\displaystyle{t}=\sqrt{{3}}$$