1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
jok3333 [9.3K]
3 years ago
13

At time t=0, a particle is located at the point (3,6,9). It travels in a straight line to the point (5,2,7), has speed 8 at (3,6

,9) and constant acceleration 2i−4j−2k. Find an equation for the position vector of the particle.
Physics
1 answer:
Elis [28]3 years ago
4 0

The particle has constant acceleration according to

\vec a(t)=2\,\vec\imath-4\,\vec\jmath-2\,\vec k

Its velocity at time t is

\displaystyle\vec v(t)=\vec v(0)+\int_0^t\vec a(u)\,\mathrm du

\vec v(t)=\vec v(0)+(2\,\vec\imath-4\,\vec\jmath-2\,\vec k)t

\vec v(t)=(v_{0x}+2t)\,\vec\imath+(v_{0y}-4t)\,\vec\jmath+(v_{0z}-2t)\,\vec k

Then the particle has position at time t according to

\displaystyle\vec r(t)=\vec r(0)+\int_0^t\vec v(u)\,\mathrm du

\vec r(t)=(3+v_{0x}t+t^2)\,\vec\imath+(6+v_{0y}t-2t^2)\,\vec\jmath+(9+v_{0z}t-t^2)\,\vec k

At at the point (3, 6, 9), i.e. when t=0, it has speed 8, so that

\|\vec v(0)\|=8\iff{v_{0x}}^2+{v_{0y}}^2+{v_{0z}}^2=64

We know that at some time t=T, the particle is at the point (5, 2, 7), which tells us

\begin{cases}3+v_{0x}T+T^2=5\\6+v_{0y}T-2T^2=2\\9+v_{0z}T-T^2=7\end{cases}\implies\begin{cases}v_{0x}=\dfrac{2-T^2}T\\\\v_{0y}=\dfrac{2T^2-4}T\\\\v_{0z}=\dfrac{T^2-2}T\end{cases}

and in particular we see that

v_{0y}=-2v_{0x}

and

v_{0z}=-v_{0x}

Then

{v_{0x}}^2+(-2v_{0x})^2+(-v_{0x})^2=6{v_{0x}}^2=64\implies v_{0x}=\pm\dfrac{4\sqrt6}3

\implies v_{0y}=\mp\dfrac{8\sqrt6}3

\implies v_{0z}=\mp\dfrac{4\sqrt6}3

That is, there are two possible initial velocities for which the particle can travel between (3, 6, 9) and (5, 2, 7) with the given acceleration vector and given that it starts with a speed of 8. Then there are two possible solutions for its position vector; one of them is

\vec r(t)=\left(3+\dfrac{4\sqrt6}3t+t^2\right)\,\vec\imath+\left(6-\dfrac{8\sqrt6}3t-2t^2\right)\,\vec\jmath+\left(9-\dfrac{4\sqrt6}3t-t^2\right)\,\vec k

You might be interested in
3<br><br><br> help with physics!!<br><br> ----------------<br><br> will give brainliest
Zinaida [17]

Answer:

Answer B

Explanation:

An increase in resistance makes it harder for the electric current to pass through

6 0
2 years ago
When an object is at a distance of twice the focal length from a concave lens, the image produced is virtual and smaller than th
Elenna [48]
The right answer for the question that is being asked and shown above is that: "The image produced is virtual and of the same size as the object." the image if the object is shifted closer to the lens to a point one focal length away from it is that The image produced is virtual and of the same size as the object.<span>
</span>
3 0
3 years ago
Read 2 more answers
When light passes through an object unchanged, scientists call that process _____.
Nataly [62]

<u>Answer</u>:

When light passes through an object unchanged, scientists call that process Transmission.

<u>Explanation</u>:

Transmission is the process where all the light that is passed through the material moves via the material without being absorbed. The Transmission depends on the affected radiation.The Transmittance of the medium is defined as the ratio between transmitted radiant power and incident radiant power. The light that is passed through the medium and not reflected will be either scattered or reflected. The light can be transmitted only through transparent or translucent material. Opaque object does not allows transmission of light.

5 0
3 years ago
Why is it important to be able to trace the pole connection on a meter back to the same type of pole at the electrical source?
lozanna [386]

Answer:

Explanation:

A grounded wire is sometimes strung along the tops of the towers to provide lightning protection.

In areas where the neutral is grounded or earthed, it is essential to endure that the neutral and the live or hot wires are not confused for each other.

When this happens, the fuses on the transformer will not operate unless the fault is very close to the transformer. The fuses in the consumer's intake box, will not operate.

6 0
3 years ago
Instrument used to measure capacity​
Andre45 [30]

Answer:

measuring cup

The most popular tool used to measure capacity is the measuring cup.

6 0
3 years ago
Read 2 more answers
Other questions:
  • What is an organism's specific role in an ecosystem
    9·2 answers
  • In a thermostat, what property of the bimetallic coil allows it to contract and expand? The two metals absorb different amounts
    10·2 answers
  • Explain how a volcanologist can piece together the history of a volcano by studying the rock that makes up the volcano
    10·1 answer
  • How are frequency and period related to each other? A. They are the same for any given wave B. They have the same magnitude but
    8·1 answer
  • Which best describes the speed of the ball as it is thrown straight up into the air and comes back down
    7·1 answer
  • Demonstrate the path light takes when traveling through the convex lens. Include an explanation of the drawing.
    11·1 answer
  • Mt. Asama, Japan, is an active volcano complex. In 2009, an eruption threw solid volcanic rocks that landed far from the crater.
    10·1 answer
  • Hi:) what does the one highlighted in pink mean?
    14·1 answer
  • True or false
    9·1 answer
  • Why are there more fat, shaggy bison in Time 3 than in Time 1?
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!