2D Particle Motion: Coordinate Systems

In two-dimensional motion analysis, which type of coordinate system is characterized by radial distance and angle?

Which of the following coordinate systems involves describing the motion of a particle along a curved path using coordinates that are tangent and normal to the path?

If the acceleration of a particle is given as a function of time, what mathematical operation is used to determine the velocity of the particle?

Given a particle moving in a plane with acceleration (\vec{f} = 3t \hat{i} + 2 \hat{j}), which expression represents its velocity (\vec{v}) if the initial velocity at $t=0$ is (\vec{v_0} = 3\hat{i} - 2\hat{j})?

A particle's motion is described in Cartesian coordinates with its position vector given by (\vec{r} = x\hat{i} + y\hat{j}). If (\hat{i}) and (\hat{j}) are unit vectors along the x and y axes, respectively, what do 'x' and 'y' represent?

Given the velocity components of a particle as functions of time, what mathematical operation determines the displacement of the particle?

A particle starts from position ((3, -2)) with an initial velocity of (3\hat{i} - 2\hat{j}) and is subjected to a constant acceleration of (3\hat{i} + 2\hat{j}). What is the position vector (\vec{r}) of the particle at time (t)?

What are the unit vectors in the directions of increasing (r) and (\theta) in a polar coordinate system?

Given (\hat{r} = \cos \theta \hat{i} + \sin \theta \hat{j}) and (\hat{\theta} = -\sin \theta \hat{i} + \cos \theta \hat{j}), what is the time derivative of (\hat{r}), denoted as (\dot{\hat{r}})?

In polar coordinates, if the position vector is given by (\vec{r} = r \hat{r}), what is the expression for the velocity vector (\vec{v})?

What are the radial and transverse components of acceleration in polar coordinates?

In central force motion, what quantity is conserved?

If a particle moves in a plane such that its speed components are constant, and one component is along a fixed line OX, while the other is perpendicular to OP (where O is the origin and P is the particle's position), what can be said about the force acting on the particle?

A particle is moving under a central force F, and its motion is described by the equations (m(\ddot{r} - r\dot{\theta}^2) = -F) and (\frac{d}{dt}(r^2\dot{\theta}) = 0). What does the second equation imply?

What is the differential equation that describes the path of a particle under a central force, where (u = 1/r)?

What physical quantity does the constant (h) represent in the analysis of central force motion, and how is it related to the particle's motion?

What condition defines an apsidal point in a central force orbit?

What is the term for the angle between two consecutive apsidal points in a central force orbit?

If a particle moves under a central force that results in a circular orbit, what can be said about the speed of the particle?

A particle is projected from an apsidal distance (a) with a speed equal to (\sqrt{2}) times the speed for a circular orbit of radius (a). What is the equation of the path?

What general equation relates the force to u in a central orbit?

What shape of orbit would one expect if (n = 1)?

What formula is used to describe the tangent made as the distance between a curve and a vector nears zero?

In what direction would there be zero traverse acceleration?

What is the simplified APSE distance equation?

What formula would express the relationship between a wire on the horizontal plane to a vertical wire?

If a particle can experience a force (\mu u^3) what equation would one use?

If angular momentum is conserved can it only move in one way?

If considering the force on an object, what direction would also need considering?