# For a short time the position of a roller-coaster car along its path is defined by the equations r=25 m, θ=(0.3t) rad, and z=(−8 cosθ) m, where t is measured in seconds, Determine the magnitudes of the car's velocity and acceleration when t=4s .

Velocity = v = 2.24m/s

Acceleration = a = 0.20m/s²

Explanation:

Please see attachment below.

Given

z=(−8 cosθ) and θ = 0.3t

z = -8Cos (0.3t)

V = dz/dt

a = v²/R.

Please see full solution below.

The roller coaster's velocity and acceleration at t=4 seconds is 7.64 m/s and 0.57 m/s² respectively.

### Explanation:

The question is about understanding kinematics in cylindrical coordinates to analyze the motion of a roller coaster car. First, we need to understand that in polar coordinates, θ is changing with time t. So, the velocity vector v will have two components, one in the θ direction (rθ') and another in the z direction (z'). Given θ = 0.3t, we differentiate θ with respect to time to get θ' or dθ/dt = 0.3 rad/sec. Then, the z component of the velocity can be calculated by differentiating the equation of motion in the z-direction, z = -8 cos(θ), with respect to time. This gives z' = 8(0.3)sin(0.3t). So, at t=4s, z' = 8(0.3)sin(1.2) = 1.89 m/s. We then calculate rθ' = r*dθ/dt = 25*0.3 = 7.5 m/s.

The magnitude of velocity can then be calculated using the Pythagorean theorem: √((rθ')² + (z')²) = √((7.5)² + (1.89)²) = 7.64 m/s .

In a similar way, we can find the acceleration components. Given that r=25 m and is constant, radial acceleration is zero ( ar = r*(θ')²). The tangential acceleration is at = r*θ'' = r*d²θ/dt² =0 m/s² and z'' = dz'/dt = 8*0.3²*cos(0.3t). So, at t = 4s, z'' = 8(0.09)cos(1.2) = 0.57 m/s². The magnitude of the acceleration is given by √((ar)² + (at)² +(z'')²) = √((0)² + (0)² +(0.57)²)= 0.57 m/s².

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## Related Questions

3. A particle of charge +7.5 µC is released from rest at the point x = 60 cm on an x-axis. The particle begins to move due to the presence of a charge ???? that remains fixed at the origin. What is the kinetic energy of the particle at the instant it has moved 40 cm if a) ???? = +20 µC and b) ???? = −20 µC?

HSBC keen vs kg get it yyyyyuuy

Explanation:

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A crate of eggs is located in the middle of the flatbed of a pickup truck as the truck negotiates a curve in the flat road. The curve may be regarded as an arc of a circle of radius 35.0 m. If the coefficient of static friction between crate and truck is 0.600, how fast can the truck be moving without the crate sliding?

v = 14.35 m/s

Explanation:

As we know that crate is placed on rough bed

so here when pickup will take a turn around a circle then in that case the friction force on the crate will provide the necessary centripetal force on the crate

So here we have

here we have

now we know that

here we have

R = 35 m

g = 9.81 m/s/s

now plug in all values in above equation

The electrons in the beam of a television tube have an energy of 19.0 keV. The tube is oriented so that the electrons move horizontally from north to south. At the electron's latitude the vertical component of the Earth's magnetic field points down with a magnitude of 42.3 μT. What is the direction of the force on the electrons due to this component of the magnetic field?

The direction is due south

Explanation:

From the question we are told that

The energy of the electron is

The earths magnetic field is

Generally the force on the electron is perpendicular to the velocity of the elecrton and the magnetic field and this is mathematically reresented as

On the first uploaded image is an  illustration of the movement of the electron

Looking at the diagram  we can see that in terms of direction  the magnetic force  is

generally  i cross k = -j

so the equation above becomes

This show that the direction is towards the south

To calculate the change in kinetic energy, you must know the force as a function of _______. The work done by the force causes the kinetic energy change.'

Answer: The force is as a function of Distance

Explanation:

The force and distance must be parallel to each other. Only the component of the force in the same direction as the distance traveled does any work. Hence, if a force applied is perpendicular to the distance traveled, no work is done. The equation becomes force times distance times the cosine of the angle between them.

where both the force F and acceleration are vectors. This makes sense since both force and acceleration have a direction.

On the other hand, the kinetic energy

K=12mv2

looks completely different. It doesn't seem to depend on the direction.

Distance

Explanation:

dW = F. dx

The volume control on a stereo is designed so that three clicks of the dial increase the output by 10 dB. How many clicks are required to increase the power output of the loudspeakers by a factor of 100?

300 clicks...

Explanation:

Output on 3 clicks = 10 dB

Increasing 10 by a factor of 100 equals 1000 dB so,

Its simple math, clicks will also increase in the same ratio and it shall take 300 clicks to increase the volume by a factor of 100.

An object of mass m moves to the right with a speed v. It collides head-on with an object of mass 3m moving with speed v/3 in the opposite direction. If the two objects stick together, what is the speed of the combined object, of mass 4m, after the collision

The speed of the combined object after collision is 0 m/s.

Total momentum before collision = total momentum after collision

m₁u₁ + m₂u₂ = (m₁ + m₂)a

m₁ = object 1 mass = m, u₁ = velocity of object 1 before collision = v, m₂ = mass of object 2 = 3m, u₂ = velocity of object 2 before collision = -v/3, a = velocity after collision

mv + 3m(-v/3) = (m + 3m)a

mv - mv = 4ma

0 = 4ma

a = 0 m/s

The speed of the combined object after collision is 0 m/s.

Find out more at: brainly.com/question/17166755

the answer is 0 m/s

Explanation:

This question is describing the law of conservation of momentum

First object has mass =m

velocity of first object = v

second object = 3m

velocity of second object = v/3

the law of conservation of momentum is expressed as

m1V1 - m2V2 = (m1+ m2) V

substituting the parameters given;

making V as the subject of formular

V =

V =

V =

= 0 m/s