Questions tagged [kinematics]

For questions on the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the mass of each or the forces that caused the motion.

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What function can I use in order to represent the position of an airplane during take-off?

I am currently doing an IA in IB maths on how to model the takeoff of an airplane. However, I did not find any equation to represent the takeoff position of the airplane. I also do not know if I have ...
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Is there an easier way to derive this equation for this mechanics problem?

We're given a semicircle whose total mass $M$ is evenly distributed along its contour and a little particle of mass $m$ is dropped from its top left corner as in the image below all over an horizontal ...
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The material point M (m=1-mass) in the Oxy plane moves on the effect of a force field which components are Fx=y,Fy=x.

Can someone help me with this exercise? The material point M (m=1) in the Oxy plane moves on the effect of a force field which components are $F_{x}=y$,$F_{y}=x$. Is there a power function and if so, ...
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Inverse Kinematic of a sun tracker with swaped rotation axis

Hi I am building a sun tracker system like the next image: Basically i need to solve: (MotorAngle_1,MotorAngle_2)= F_Inverse_Kinematic(sun_azimuth,sun_elevation) Where sun_azimuth and sun_elevation: ...
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When it's better choose an specific Dx than calculate the Derivative?

I was thinking about this problem: a car that moves as x(t) = t^3 in t=0 if i calculate the derivative in that point is 0. But in the reality the car near the time 0 its accelerating. If i choose ...
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2 votes
2 answers
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maximum range of projectile launched from elevation

I am trying to find the maximum range of projectile from an elevation. I found the answer in this question, but I have two questions: Why does $y$ need to be $0?$ Why do they differentiate with ...
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Finding the location of a particle given angular acceleration and linear acceleration

Given a rocket with known initial position, linear velocity, linear acceleration, orientation, angular velocity, and angular acceleration, and knowing that the linear acceleration will always be in ...
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How to solve equation for drag force?

I've been working on graphing the movements of a sphere when thrown. I'm trying to simulate air resistance. I've found the equation $$F_d=\frac12 C_d\rho V^2A$$ I've separated the velocity into x and ...
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Horizontal SUVAT problem (AS Level)

Could someone please explain how you get the answer for the following question: ...
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Physical model: how do you solve for a variable in the argument of multiple trigonometric functions?

I was modeling a situation I've come across before, and derived an equation for the solution. WolframAlpha calculated the solution, but gave no explanation as to how it did it. I'll give the equation ...
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Find the velocity of the falling object when it touches the water

The question: An object falls from a hovering surf-lifesaving helicopter over a port at $500$ m above sea level. Find the velocity of the object when it hits the water when the acceleration of the ...
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Variable acceleration problem

So here is the question: A particle moves in a straight line. When the particle’s displacement from a fixed origin is $x$ m, its velocity is $v$ m/s and its acceleration is $a$ m/$s^2$. Given that $a =...
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2D joystick coordinates to 3D cartesian coordinates

I am reading XY values from a joystick, such as this one. The joystick provides 0 to 1023 values in both x and y directions, but I have scaled the range to be between -512 and 512, thus, the center ...
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2 answers
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Generalizing one motion equation

So, in physics we have a pretty common equation of motion which is: $$v_f^2 = v_0^2 + 2a(s_f-s_0)$$ which is valid for one-dimensional movement with constant acceleration. We can generalize it for any ...
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Non-linear initial value problem

Problem Consider the following Initial Value Problem (IVP) \begin{equation*} \begin{aligned} \dot{\xi}(t) &= v(t)\,\cos[h(t)]\\ \dot{v}(t) &= a_{\parallel}\\ \dot{h}(t) &=\omega \end{...
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Suppose the equation of motion of a material point is given by the equation $s(t)=e^t$. Does it follow that $s(t)=v(t)=a(t)=e^t$?

Suppose that the path traveled by a material point changes according to the law $s(t)=e^t$, where $e$ is Euler's number. We know that acceleration is the first derivative of velocity and the second ...
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proof that the instantaneous center of rotation has to have zero speed

We're on the plane $\mathbb C$ or $\mathbb R^2$. Definition 1: A movement is a colection of isometries $H_t$ in the plane, with $t \in [0,1]$ and $H_0$ being the identity isometry and such that $H_t(X)...
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Trajectory of a 2D constant jerk motion

This thread is a natural prosecution of this other. Problem Let $\xi,\eta$ be the horizontal and vertical coordinates of a plane. Consider the following sequence of point \begin{equation}\begin{...
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Cartesian equation of uniformly accelerated motion

Problem Let $\xi,\eta$ be the horizontal and vertical coordinates of a plane. Consider the following sequence of point \begin{equation}\begin{aligned} \xi_k &= \xi_0 + kT \dot{\xi}_0 + k^2 \frac{T^...
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Can anyone help me solve this hard physics vector problem? Thanks

River vector problem John is on the west bank of a river that flows from north to south at a speed of $5 \text{m/s}$. She is easily able to swim at a constant speed of $1\text{m/s}$. The river is $30 \...
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Problem following derivation of radius of curvature

I don't understand where the expression in the box is coming from. Can anyone please help me? I can follow up to the equality to the left of the box but i don't see how - dy/d^2x can be equal to that!
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The cat and mouse problem

My professor has given me an extremely hard physics problem I need the solution to. Basically the problem is as follows:a cat is sitting in the bottom left corner of a room with sides length a,while a ...
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Calculus and kinematics problem

The tunnelling speed of a machine is shown is modelled as $$v=\frac{k}{s}$$ where k is the constant with the units $m^2/hr$ and $s$ is the length of the tunnel already dugout. I am to find a ...
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Overtaking a moving object in hyperbolic geometry.

Suppose that a sailboat $S$ moves in a straight line (geodesic) with constant speed $v$ in an infinite hyperbolic sea. You try to overtake that sailboat with a motorboat $M$ which has maximal speed $u$...
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Finding uncertainty in acceleration given uncertainty in position

I used LoggerPro to analyze a video of the motion of a falling chained object. From LoggerPro, I found position-time data, modeled it into an exponential function, and found its second derivative to ...
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1 answer
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How to solve the inverse kinematics of a 2DOF system on a rotating platform?

I have a 2DOF (z,y axes) stabilization system that needs to maintain the orientation of the end-effector. Suppose the system is attached to a 3DOF rotating platform and has some fixed joints at the ...
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3 votes
1 answer
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How to find the rotation matrix (with no x rotation) between two rotation matrices?

I need to find the rotation matrix (with no $x$ rotation) between two rotation matrices. Given a starting rotation matrix $\textbf{R}_a$ and a setpoint $\textbf{R}_{SP}$. I need to find the rotation ...
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Drive unicycle kinematics equations through the geometry

Consider a unicycle model as described here. The unicycle dynamics in discrete-time can be written as: $$ \begin{array}{l} {x_{k + 1}} = {x_k} + \Delta {s_{k + 1}} \times \cos ({\theta _{k + 1}})\\ {...
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Calculate time to collision?

I have two objects ($a$ and $b$) moving towards each other at different speeds ($30$mph and $50mph$ respectively). The distance ($d$) between them is, let's say 2 miles. I initially came across ...
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Pseudo Inverse of Jacobian Matrix

I have a script written by someone else that outputs end-effector velocities. I need to transform these end-effector velocities to joint velocities. This requires the pseudo-inverse of the Jacobian ...
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6 votes
1 answer
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Integrating product of Dirac deltas and step functions

I have the following integral $$\int d^4\boldsymbol{x}' \,\delta\big[(\boldsymbol{x}-\boldsymbol{x'})^2+\alpha^2\big]\,\Theta(-x_0+x'_0)\,\delta\big[(\boldsymbol{x'})^2+\alpha^2\big]\,\Theta(-x_0'),\...
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Find unit normal vector given speed-, acceleration- and jerk vectors, Calculus III

So we have been given the following: $$\frac{dr}{dt} = (-3,2,0)$$ $$\frac{d^2r}{dt^2} = (0,3,-3)$$ $$\frac{d^3r}{dt^3} = (0,0,1)$$ With the information above, I have found the unit tangent vector by ...
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Why does substitution yield a different result to integration? (Kinematics Example)

Method 1: Let $x=t^2$, find a general solution of $v$ in terms of $x$. (Kinematics, $v=x'(t)$) $v = 2t$ --> $t = \frac{v}{2}$ so, $x=\frac{v^2}{4}$ Method 2: $a = x''(t) = 2$ $v*\frac{dv}{dx} = 2$ $...
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Calculation of the induced rotation angle between two moving bodies

I'm needing your help to determine the rotation angle between two moving points A and K (i estimate that there is an induced rotation of A around K with respect to the y axis in the attached frame) ...
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Need help to verify the steps to my Inverse Kinematics solution

I am trying to implement the Jacobian IK solution in my program. I have a single kinematic chain with N rotational or transnational joints. The implementation is independent of configuration (i.e I ...
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Can I estimate $\dot{\omega}$ in this equation, even though I cannot directly solve the equation to it?

I am trying to measure/estimate the angular acceleration of an object $\dot{\omega} $ from a measurement of it's acceleration (using an accelerometer) $^{i} {\boldsymbol{a}}_m$. As far as I understand,...
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1 answer
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Angle of rotation of the bar when sphere is hitting it tangentially

A rectangular bar is pivoted at point P, the bar can rotate freely about this pivot point P. A ball with radius $R$ (here shown as a circle for simplicity) can only move/slide in the x-axis direction. ...
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Does Euler's rotation theorem assume a continuous movement over time?

Or can we apply ANY isometry in a sphere and it will have a fixed diameter? Regardless if it is discontinuous in time. Would absolutely any discontinuous isometry be a rotation in a sphere?
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Determine if a value falls between two limits when the upper limit can become lower than the lower limit

I have a rotary device that provides feedback in the range of 0-359.99 degrees, once the device moves past 359.99 the feedback restarts at 0. I command the device to a degree position and then I ...
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3DOF Stewart Platform Algorithm

I am currently building a 3DOF Stewart Platform which uses linear actuators to create the tilt, pitch, and roll angles. Is there an algorithm I can use to calculate the displacement of each actuator ...
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Inverse kinematics algebraic solutions: Multiplying each side of transform equation by an inverse

I am looking at closed form algebraic solutions to inverse kinematics problems (in this case for robot manipulators). The correct solution found the transform matrices via forward kinematics (using ...
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Angle between two Quaternions in x or y

I'am working with Quaternion, I have two Quaternions, first is an initiale Quaternion, and I have a second Quaternion, I'd like to detect if my second Quaternion has rotate more than 90° in x or y is ...
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Determine the third coordinate of a Right triangle for the given three sides and other two coordinates of triangle in 3D space?

I am dealing with marker data in 3D space. I have a right triangle made by the three points [A, B, C] where AC = Opposite, BC = Hypotenuse, AB = Adjacent. The objective is to find the coordinates of ...
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4 answers
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Deriving Kinematics Equation without Calculus / First Principle

How do I derive the Kinematics equation $v=v_0+at$ from the equation $x = x_0+v_0t+\frac12at^2$? This is how I have tried : $$x - x_0=v_0t+\frac12at^2$$ $$\Delta x= t(v_0+\frac12at)$$ $$\dfrac{\Delta ...
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Reconstructed rotation matrix is not the same as the original one

I need a little help. I rotated the vector $s$ using the rotation matrix $R$ and got a new vector $S$. Now I want to reconstruct the rotation matrix from vectors $s$ and $S$ using the Rodrigues ...
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Projectile Motion - Trajectories From Points

For a little background, this problem came up due to something I'm trying to code, however the problem itself is disparate from that. I know the basics of projectile motion and kinematics from a pair ...
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Get a 3d Pose from Acceleration + rotation angles

I am trying to get a pose (x,y,z + angles x,y,z). What is given (i is a discrete time t_i): pos_x(i), pos_z(i), pos_z(i) rot_x(i), rot_y(i), rot_z(i) a_x(i), a_y(i), a_z(i) a_rot_x(i), a_rot_y(i), ...
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4 votes
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Time derivative of a constant vector in rotating frame

Recently I'm learning rotating frame kinematics, but I have come up with a question that confuses me a lot. Let's say a vector $P$ connects the origin and a fixed point in the rotating frame, so it is ...
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How to find $x$ and $y$ components of a vector if angle between two vectors is Obtuse (greater than 90)?

How to divide $Q$ and $P$ into its components? Can you write $θ$ in terms of P and Q. if yes then how to write it and from which equation you are calculating the θ?
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2 votes
2 answers
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How to calculate distance of shortest path when there is a flow in the river?

I am trying to solve this question: I have figured out a solution involving calculus, however there seems to be a much simpler solution that was provided with the question, as follows: However, I ...
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