공간에서의 운동

The force on a particle is given by $\text{f}(t)=(\cos t)\text{i}+(\sin t)\text{j}.$ The particle is located at point $(c,0)$ at $t=0.$ The initial velocity of the particle is given by $\text{v}(0)={v}_{0}\text{j}.$ Find the path of the particle of mass m. (Recall, $\text{F}=m\cdot \text{a}.)$

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?

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.

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.

Show that the particle moves on a circular cone.

Find the angle between the velocity and acceleration vectors when $t=1.5.$

Find the tangential and normal components of acceleration when $t=1.5.$

How is velocity different from speed?

What is the difference between displacement and distance traveled?

What is the difference between average velocity and average speed?

Distance traveled is the same as _____ _____, just viewed in a different context.

Describe a scenario where an object's average speed is a large number, but the magnitude of the average velocity is not a large number.

Explain why it is not possible to have an average velocity with a large magnitude but a small average speed.

In Exercises 7–10 , a position function $\vec r(t)$ is given. Find $\vec v(t)$ and $\vec a(t)$.

$\vec r(t) = \left\langle 2t+1, 5t-2, 7\right\rangle$

In Exercises 7–10 , a position function $\vec r(t)$ is given. Find $\vec v(t)$ and $\vec a(t)$.

$\vec r(t) = \left\langle 3t^2-2t+1, -t^2+t+14\right\rangle$

In Exercises 7–10 , a position function $\vec r(t)$ is given. Find $\vec v(t)$ and $\vec a(t)$.

$\vec r(t) = \left\langle \cos t,\sin t\right\rangle$

In Exercises 7–10 , a position function $\vec r(t)$ is given. Find $\vec v(t)$ and $\vec a(t)$.

$\vec r(t) = \left\langle t/10,-\cos t,\sin t\right\rangle$

In Exercises 11–14 , a position function $\vec r(t)$ is given. Sketch $\vec r(t)$ on the indicated interval. Find $\vec v(t)$ and $\vec a(t)$, then add $\vec v(t_0)$ and $\vec a(t_0)$ to your sketch, with their initial points at $\vec r(t_0)$, for the given value of $t_0$.

$\displaystyle \vec r(t) = \left\langle t,\sin t \right\rangle$ on $[0,\pi/2]$; $t_0= \pi/4$

In Exercises 11–14 , a position function $\vec r(t)$ is given. Sketch $\vec r(t)$ on the indicated interval. Find $\vec v(t)$ and $\vec a(t)$, then add $\vec v(t_0)$ and $\vec a(t_0)$ to your sketch, with their initial points at $\vec r(t_0)$, for the given value of $t_0$.

$\displaystyle \vec r(t) = \left\langle t^2,\sin t^2 \right\rangle$ on $[0,\pi/2]$; $t_0=\sqrt{\pi/4}$

In Exercises 11–14 , a position function $\vec r(t)$ is given. Sketch $\vec r(t)$ on the indicated interval. Find $\vec v(t)$ and $\vec a(t)$, then add $\vec v(t_0)$ and $\vec a(t_0)$ to your sketch, with their initial points at $\vec r(t_0)$, for the given value of $t_0$.

$\displaystyle \vec r(t) = \left\langle t^2+t,-t^2+2t \right\rangle$ on $[-2,2]$; $t_0=1$