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
babunello [35]
3 years ago
10

If position vector r = bt^2i + ct^3j, where b and c are positive constants, when does the velocity vector make an angle of 450 w

ith the x-and y-axes?
Physics
1 answer:
vazorg [7]3 years ago
7 0

\vec r(t)=bt^2\,\vec\imath+ct^3\,\vec\jmath

The velocity at time t is

\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=2bt\,\vec\imath+3ct^2\,\vec\jmath

Take two vectors that point in the positive x and positive y directions, such as \vec\imath and \vec\jmath. The dot products of the velocity vector with \vec\imath and \vec\jmath are

\dfrac{\mathrm d\vec r(t)}{\mathrm dt}\cdot\vec\imath=2bt=\sqrt{4b^2t^2+9c^2t^4}\cos\theta

and

\dfrac{\mathrm d\vec r(t)}{\mathrm dt}\cdot\vec\jmath=3ct^2=\sqrt{4b^2t^2+9c^2t^4}\cos\theta

We want the angles between these vectors to be 45º, for which we have \cos45^\circ=\frac1{\sqrt2}. So

\begin{cases}2\sqrt2\,bt=\sqrt{4b^2t^2+9c^2t^4}\\3\sqrt2\,ct^2=\sqrt{4b^2t^2+9c^2t^4}\end{cases}\implies3\sqrt2\,ct^2-2\sqrt2\,bt=0

\implies t(3ct-2b)=0

\implies t=0\text{ or }t=\dfrac{2b}{3c}

When t=0, the velocity vector is equal to the zero vector, which technically has no direction/doesn't make an angle with any other vector. So the only time this happens is for

\boxed{t=\dfrac{2b}{3c}}

You might be interested in
The sound intensity of a certain type of food processor in normally distributed with standard deviation of 2.9 decibels. If the
Maru [420]

Answer: (48.41,\ 52.19)

Explanation:

The confidence interval for population mean is given by :-

\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}

Given : Sample size : n=9

Sample mean : \ovreline{x}=50.3\text{ decibels}

Standard deviation : \sigma=2.9\text{ decibels }

Significance level : \alpha=1-0.95=0.05

Critical value : z_{\alpha/2}=z_{0.025}=1.96

Now, the 95% confidence interval estimate of the (true, unknown) mean sound intensity of all food processors of this type :-

50.3\pm (1.96)\dfrac{2.9}{\sqrt{9}}\\\\\approx50.3\pm1.89\\\\=(50.3-1.89,\ 50.3+1.89)=(48.41,\ 52.19)

6 0
3 years ago
2) A constant net force acts on an object. Describe the motion of the object.?
77julia77 [94]
Constant acceleration
7 0
3 years ago
The Coefficient of kinetic friction between the tires of your car and the roadway is \"μ\". (a) If your initial speed is \"v\" a
vazorg [7]
We make use of the equation: v^2=v0^2+2a Δd. We substitute v^2 equals to zero since the final state is halting the truck. Hence we get the equation           -<span>v0^2/2a = Δd. F = m a from the second law of motion. Rearranging, a = F/m
</span>F = μ Fn where the force to stop the truck is the force perpendicular or normal force multiplied by the static coefficient of friction. We substitute, -v0^2/2<span>μ Fn/m</span> = Δd. This is equal to 
3 0
3 years ago
Read 2 more answers
Which classification scheme is based on the brightness of a star as it appears to you in the night sky?
anastassius [24]
The apparent magnitude scale is a classification scheme which is based on the brightness of stars. The range of brightness values is from 1 to 6.
The stars which are the most brightest are ranked as number 1 and also called first magnitude stars, stars which are little dimmer than number 1 are ranked as number 2 and also called second magnitude stars. Similarly the most faintest stars are ranked number 6 and also called as the sixth magnitude stars.
3 0
3 years ago
Which statement is true about acceleration?
Ilia_Sergeevich [38]
The answer is C is think
8 0
3 years ago
Other questions:
  • 20pts! What are the thin fibers of the human nervous system called?
    7·2 answers
  • If the amount of gas in a sealed container increases, what happens to the pressure in the container
    10·2 answers
  • A ball of gas becomes a ____ when nuclear fusion begins in its core.
    11·1 answer
  • Solve the equation 3x^2+8x+5=0
    5·1 answer
  • Lidia plans an experimental investigation to see how the thickness of a lens affects the point where a beam of light is focused.
    8·1 answer
  • Suppose the entire solar nebula had cooled to 50 K before the solar wind cleared the early solar system of its gases. How would
    13·1 answer
  • One horsepower is equal to how many watts
    14·1 answer
  • The velocity of a moving body increases from 10m/s to 15 m/s in 5 sec . calculate its acceleration ​
    14·1 answer
  • What can i do to recycle?
    9·1 answer
  • - Find the velocity of a wave in the ocean if the wavelength is 2.0m and the frequency is 0.56Hz.
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!