# Solving equations of motion Given an acceleration vector, initial velocity langle u_0, v_0, w_0rangle, and initial position langle x_0, y_0, z_0rangle, find the velocity and position vectors, for t >= 0. a(t) = langle1, t, 4trangle, langle u_0, v_0, w_0rangle = langle20, 0, 0rangle, langle x_0, y_0, z_0rangle = langle0, 0, 0rangle

Question
Equations and inequalities
Solving equations of motion Given an acceleration vector, initial velocity $$\displaystyle{\left\langle{u}_{{0}},{v}_{{0}},{w}_{{0}}\right\rangle}$$, and initial position $$\displaystyle{\left\langle{x}_{{0}},{y}_{{0}},{z}_{{0}}\right\rangle}$$, find the velocity and position vectors, for $$\displaystyle{t}\ge{0}$$.
$$\displaystyle{a}{\left({t}\right)}={\left\langle{1},{t},{4}{t}\right\rangle},{\left\langle{u}_{{0}},{v}_{{0}},{w}_{{0}}\right\rangle}={\left\langle{20},{0},{0}\right\rangle},{\left\langle{x}_{{0}},{y}_{{0}},{z}_{{0}}\right\rangle}={\left\langle{0},{0},{0}\right\rangle}$$

2021-01-03

### Relevant Questions

solving equation of motion
Given an accelaration vector,initial velocity $$\displaystyle{\left\langle{0},{10}\right\rangle}$$,and initial position $$\displaystyle{\left\langle{x}_{{{0}}},{y}_{{{0}}}\right\rangle}$$,find the velocity and position vectors $$\displaystyle{t}\geq{0}$$
$$\displaystyle{a}{\left({t}\right)}={\left\langle{0},{10}\right\rangle},{\left\langle{u}_{{{0}}},{v}_{{{0}}}\right\rangle}={\left\langle{0},{5}\right\rangle},{\left\langle{x}_{{{0}}},{y}_{{{0}}}\right\rangle}={\left\langle{1},-{1}\right\rangle}$$
Given an acceleration vector,initial velocity $$\displaystyle{\left\langle{u}_{{{0}}},{v}_{{{0}}}\right\rangle}$$ ,and initial position $$\displaystyle{\left\langle{x}_{{{0}}},{y}_{{{0}}}\right\rangle}$$,find the velocity and position vectors for $$\displaystyle{t}\geq{0}$$
$$\displaystyle{a}{\left({t}\right)}={\left\langle{0},{1}\right\rangle},{\left\langle{u}_{{{0}}},{v}_{{{0}}}\right\rangle}={\left\langle{2},{3}\right\rangle},{\left\langle{x}_{{{0}}},{y}_{{{0}}}\right\rangle}={\left\langle{0},{0}\right\rangle}$$
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
A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 8.50 N is applied. A 0.530-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x = 5.00 cm and released from rest at t = 0. (Assume that the direction of the initial displacement is positive.)
(a) What is the force constant of the spring? 280 N/m
(b) What are the angular frequency (?), the frequency, and the period of the motion?
f = 3.6817 Hz
T = 0.27161 s
(c) What is the total energy of the system? 0.35 J
(d) What is the amplitude of the motion? 5 cm
(e) What are the maximum velocity and the maximum acceleration of the particle?
$$\displaystyle{v}_{{\max}}={1.1561}\frac{{m}}{{s}}$$
$$\displaystyle{a}_{{\max}}={26.73}\frac{{m}}{{s}^{{{2}}}}$$
(f) Determine the displacement x of the particle from the equilibrium position at t = 0.500 s.
(g) Determine the velocity and acceleration of the particle when t = 0.500 s. (Indicate the direction with the sign of your answer.)
v = _________________ $$\displaystyle\frac{{m}}{{s}}$$
a = _________________ $$\displaystyle\frac{{m}}{{s}^{{{2}}}}$$
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.
(a) Find the velocity vector, speed, and acceleration vector of the object.
(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.
(a) Find the velocity vector, speed, and acceleration vector of the object.
(b) Evaluate the velocity vector and acceleration vector of the object at the given value of $$\displaystyle{t}=\sqrt{{3}}$$
A bird flies in the xy-plane with a position vector given by $$\displaystyle\vec{{{r}}}={\left(\alpha{t}-\beta{t}^{{3}}\right)}\hat{{{i}}}+\gamma{t}^{{2}}\hat{{{j}}}$$, with $$\displaystyle\alpha={2.4}\ \frac{{m}}{{s}},\beta={1.6}\ \frac{{m}}{{s}^{{3}}}$$ and $$\displaystyle\gamma={4.0}\ \frac{{m}}{{s}^{{2}}}$$. The positive y-direction is vertically upward. At the bird is at the origin.
Calculate the velocity vector of the bird as a function of time.
Calculate the acceleration vector of the bird as a function oftime.
What is the bird's altitude(y-coordinate) as it flies over x=0 for the first time after ?
A rocket is launched at an angle of 53 degrees above the horizontal with an initial speed of 100 m/s. The rocket moves for 3.00 s a long its initial line of motion with an acceleration of 30.0 m/s/s. At this time, its engines fail and the rocket proceeds to move as a projectile. Find:
a) the maximum altitude reached by the rocket
b) its total time of flight
c) its horizontal range.
Solving the following equations will require you to use the quadratic formula. Solve each equation for u between $$\displaystyle{0}^{{\circ}}{\quad\text{and}\quad}{360}^{{\circ}}$$, and round your answers to the nearest tenth of a degree.
$$\displaystyle{\cos{{2}}}{u}+{\sin{{u}}}={0}$$
$$x(t) = t^{3} - 6t^{2} + 9t - 9,$$
$$0 \leq t \leq 10$$
$$x'(t) = ?$$
$$x"(t) = ?$$