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
Infer why the output force exerted by a rake must be less than input force?
adell [148]
<h3><u>Answer and Explanation</u>;</h3>
  • input force refers to the force exerted on a machine, also known as the effort, while the output force is the force machines produce or the Load. The ratio of output force to input force gives the mechanical advantage of a simple machine
  • <em><u>The output force exerted by the rake must be less than the input force because one has to use force while raking. The force used to move the rake is the input force. </u></em>
  • <em><u>The rake is not going to be able to convert all of the input force into output force, the force the rake applies to move the leaves, because of friction.</u></em>
5 0
3 years ago
Newer aircraft jet catapult systems use magnets instead of steam. The launch still takes 1.19 seconds, but the acceleration is a
Westkost [7]

Answer:

33.6 m

Explanation:

Given:

v₀ = 0 m/s

a = 47.41 m/s²

t = 1.19 s

Find: Δx

Δx = v₀ t + ½ at²

Δx = (0 m/s) (1.19 s) + ½ (47.41 m/s²) (1.19 s)²

Δx = 33.6 m

8 0
3 years ago
A4.00-kg box sits on a ramp that is inclined at 30°above the horizontal.The coefficient of kinetic friction between the box and
enot [183]

From the calculation, we can see that the force that  moves the box forward is 5.7 N.

<h3>What is frictional force?</h3>

The term frictional force refers to the force that opposes motion. Given that;

ma = ukmgcosθ + F

Let F be the forward force.

Hence;

F = ma - ukmgcosθ

F = 4 * 3 - (0.32 * 4.00 * 9.8 * sin 30)

F =  5.7 N

Learn more about frictional force:brainly.com/question/14662717

#SPJ1

3 0
2 years ago
If a skydiver weighs 800 N, what would air resistance be when he is at terminal velocity?
serg [7]
We call "terminal velocity" the constant speed of a falling body
when it is no longer accelerating.

We know that if a body is not accelerating, then the net force
on it is zero.

From the question, we know that the downward force of gravity
on the skydiver is 800 N.

If the 800 N downward plus the air resistance upward add up
to zero, then the air resistance upward must also be 800 N.
5 0
3 years ago
Read 2 more answers
A car moving at 10 m/s crashes into a large bush and stops in 1.3 m. Using the Work-Energy theorem, calculate the average force
ad-work [718]

Answer:

The magnitude of the average force the seat belt exerts on the passenger is 2692.3 N.

It takes 0.26 s to bring the passenger in the car to a halt.

Explanation:

Hi there!

The negative work (W) needed to bring a moving object to stop is equal to its kinetic energy (KE):

W = KE

F · s = 1/2 · m · v²

Where:

F = applied force on the passenger.

s = displacement of the passenger.

m = mass of the passenger.

v = velocity of the passenger.

Solving the equation for F:

F = 1/2 · m · v² / s

Replacing with the data:

F = 1/2 · 70 kg · (10 m/s)² / 1.3 m

F = 2692.3 N

The magnitude of the average force the seat belt exerts on the passenger is 2692.3 N.

According to the second law of Newton:

F = m · a

Where "a" is the acceleration of the passenger.

We also know from kinematics that the velocity of an object can ve calculated as follows:

v = v0 + a · t

Where:

v = velocity of the passenger at time t.

v0 = initial velocity.

t = time.

a = acceleration.

When the passenger stops, its velocity is zero. So replacing a = F/m, let´s solve the equation for the time it takes the passenger to stop:

v = v0 + a · t

0 = 10 m/s + (-2692.3 N/ 70 kg) · t

-10 m/s / (-2692.3 N/ 70 kg)  = t

t =0.26 s

It takes 0.26 s to bring the passenger in the car to a halt.

5 0
3 years ago
Other questions:
  • A speck of dust with mass 12 mg and electric charge 10 μC is released from rest in a uniform electric field of magnitude 850 N/C
    10·1 answer
  • Some hypothetical alloy is composed of 12.5 wt% of metal A and 87.5 wt% of metal B. If the densities of metals A and B are 4.27
    8·1 answer
  • According to Newton’s first law of motion, what must happen to move an object at rest? It must move by its own inertia. It must
    10·2 answers
  • Which axis is the horizontal axis in a graph?*
    11·2 answers
  • While driving north at 21 m/s during a rainstorm you notice that the rain makes an angle of 36° with the vertical. while drivin
    6·1 answer
  • If a 0.15 kg ball falls and has a KE of 20 J just before striking the ground, from what height did it fall. A. 1.36m B. 3m C. 13
    9·1 answer
  • An automobile starter motor has an equivalent resistance of 0.055 Ï and is supplied by a 12.0 v battery which has a 0.0305 Ï int
    15·1 answer
  • Do simple sugars provide short term energy?
    13·2 answers
  • What is the velocity of a wave with a wavelength of 3.7 m and a frequency of 22.5 Hz?
    7·1 answer
  • Please help <br> i’ll give you so many points
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!