In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.
There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations, arising from the definitions of kinematic quantities: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations, rotations, oscillations, or any combinations of these.
A differential equation of motion, usually identified as some physical law and applying definitions of physical quantities, is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, which fixes the values of the constants.
To state this formally, in general an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr/dt), and its acceleration (the second derivative of r, a = d2r/dt2), and time t. Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in r is a second-order ordinary differential equation (ODE) in r,
M
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,
{\displaystyle M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,}
where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t = 0,
r
(
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{\displaystyle \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,.}
The solution r(t) to the equation of motion, with specified initial values, describes the system for all times t after t = 0. Other dynamical variables like the momentum p of the object, or quantities derived from r and p like angular momentum, can be used in place of r as the quantity to solve for from some equation of motion, although the position of the object at time t is by far the most sought-after quantity.
Sometimes, the equation will be linear and is more likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions.
s1=-u1^2/2a1
s2=-u2^2/2a2
s2>s1+d
(If distance the car stops is bigger than the distance the lorry stops plus the initial distance then they will crash)
(sub s1 and s2 in)
-u2^2/2a2 > -u1^2/2a1+d
Switch 2a2 with whole left side of equation.
-u2^2/(-u1^2/2a1)+d > 2a2
Make the (d)a fraction by...
To my understanding, suvat equations must apply when motion is one dimensional and also the acceleration is constant pointing in the direction of motion or against the direction of motion. So I'm not sure about this.
Perhaps, the vector form just means that we can select an axes system (i.e. x...
Hi
I was just wondering about the suvat formulae and a question popped into my head, which I'd like someone to try and explain the reason as to why please.
So I know that when we have a formula such as F=ma or v = u + at, you can evaluate the magnitude of both sides and arrive at a scalar...
I have calculated the acceleration of truck B from v=u+at as 5ms/s. The force applied to truck B is therefore 5x10=50N.
I am unsure whether this question is poorly worded, but I feel a reasonable assumption is that the force applied to truck A would be the same as truck B, without knowing its...
Please scroll-down to the end (Question 54): https://www.undergraduate.study.cam.ac.uk/files/publications/engineering_s1_qp_2017.pdf
I have also been referring to unofficial worked solutions (http://www.engineeringadmissionsassessment.com/2017-solutions.html), but I didn't understand how it...
Hey Everyone, my physics teacher has assigned us a task which involves predicting the range of a ball falling down an inclined plane into a free-fall, the equation for the final velocity of the ball down the ramp, accounting for rotational velocity has been provided, this is the initial velocity...
I've been attempting to solve this problem for three days now. I have thrown away my old attempts (like, scrumpled up into the bin), but my old attempts involved:
Trying to set up simultaeneous equations relating the journeys between EH and FG to find the deceleration, but the reason why this...
I'm continuing to struggle when it comes to objects that are dropped at different times.
At height h, a ball is dropped at rest.
Also at height h, another ball is dropped one second later.
For the first ball, time=t.
For the second ball, time=t+1...but this is wrong? The answers show it...
Homework Statement
A bomb is launched from the front of a plane horizontally. The inicial velocity of the bomb is ##v##, the inicial velocity of the plane is ##u##. Find:
The trayectory of the bomb with respect to the ground.The trayectory of the bombb with respect to the plane.The trayectory...
Homework Statement
I have a question asking me to find the launch speed of a ball (mass 0.39kg) when released by a spring mechanism made of 2 springs each with force constant 25Nm^-2. they are pulled back 12 cm. the ball is initially at rest.Homework Equations [/B]
v^2=u^2+2as
f=ma
f=kx...
Homework Statement
An object is released from rest in the gravitational field of the Earth. Air resistance is negligible. How far does the object move during the fourth second of its motion?
A. 15 m
B. 25 m
C. 35 m
D. 45 m
Homework Equations
v = u + at
s = ut + 1/2 at^2
v^2=u^+2as...
Homework Statement
A vehicle Accelerates from rest (a) and travels a certain distance s1 It then deccelerates (a) at a constant rate and travels a second distance s2 until it stops. If s2/s1 = K then what is the accleration (a) of the vehicle.
Homework Equations
v=u+at
s=ut+1/2at^2...
Hello. This is my first post so I apologise if I've put it in the wrong category or done something wrong.
Today in my physics lesson I was given a question where we were supposed to find the depth of a well, and we were given the suvat equations. The question was as follows:
"A stone is...
How would one calculate the drag on a projectile (in this case a 1cm3 cube) that was launched at 0°.
The vertical drop, initial velocity, distance, and time (taken to travel distance) where measured.
I want to say that I could compare these experimental drop (bellow hight that projectile was...
Hello everyone,
Please can someone help me rearrange this equation to make t the subject:
Relevant equations:
My attempt at a solution:
Thank you all for you help
Homework Statement
A golfer is about to hit a golf ball from a tee to a hole on the green. The tee and the green are at the same level. The golfer hits the ball, projecting it at a speed of 35 m/s 28° to the horizontal. Air resistance is negligible. Calculate:
a) the horizontal distance...
Homework Statement
a) How much time does a dropped weight take to fall 120m down a cliff? (I answered this correctly as 4.9s -- √(s/0.5 x g).)
b) What would the time be (in question a) if the weight was thrown downwards at 2.5m/s?
Homework Equations
SUVAT
The Attempt at a Solution
None...
Please help, I've put this is true on my personal statement, but I'm now fairly sure that it is not, but a more expert opinion would be extremely helpful.
Hi there,
Sorry in advance. This problem is very simple but I can't see where I'm going wrong or what I'm missing. It may even be a numerical mistake, although I have run the numbers a few times. I have a feeling that I'm going to kick myself but anyway...
A ball is projected vertically...
text is same as on pic.
I'm trying to express this diagram as algebraically as possible so later on I can add real world limitations into my equation and see what other requirements are needed.
I got the first and simplest one which is to be in equilibrium the mass/weight of B has to be...
Homework Statement
Object falls from a tree which is 3m above the ground, what is its velocity when it hits the ground ?
Homework Equations
mgΔh=1/2mv2
or
s=vt-1/2at2
The Attempt at a Solution
Using the idea of PE=KE, as the masses cancel out I get an answer of V=7.66ms-1
However I thought...
Attempting a projectile motion question where initial and final speed is given but angle of elevation isn't. Need to find an equation for the vertical distance H travelled. I got the right answer using v^2 = u^2 + 2as substituting s=H and a=-9.8.
However the mark scheme states that this...
Homework Statement
A ball is thrown vertically upwards from a roof of a building and it lands 3 seconds later on the ground 7 meters below the roof. Calculate:
a) the speed with which the ball was thrown upwards
b) the maximum height of the ball above the ground
c) the speed with which the ball...
Homework Statement :[/B]There is a wall of height 30 meters.From the top edge of the wall a ball is thrown up with a velocity of 5m/s.Find the time taken by the ball to come to the base of the wall.
Take g=10 m/s^2
Homework Equations :I used
V=u+at[/B]The Attempt at a Solution
v=0 as ball will...
Suvat equations -- car braking hard to avoid a collision
Homework Statement
A car (A) is speeding along a rural road, of width 4m with a speed of 25m/s when the driver sees another vehicle (B) of length 3.5m just starting to cross the road at a point 40m ahead. The drivers reaction time is...
Homework Statement
"The constant force resisting the motion of a car of mass 1500kg is 980N. If, when traveling at 48kmhr-1, the car is brought to rest in a distance 50m by applying the brakes, find the additional retarding force due to the brakes (assumed constant)
Homework Equations...
Homework Statement
"A block of mass 10kg is pulled 20m up a smooth plane inclined at 45 degrees to the horizontal. The block is initially at rest and reaches a velocity of 2.0m/s at the tope of the plane. Calculate the magnitude of the force required, assuming it acts parallel to the plane."...
I would just like someone to give a quick check to part b) of this question, just to make sure I have fully grasped the concept.
A t=0s, particle P is fired vertically upwards with speed 17.64ms-1.When P reaches its maximum height, particle Q is fired vertically upwards with a speed 25ms-1...
Homework Statement
Homework Equations
F=ma to find the respective accelerations of the man and the barrel.
The suvat equations.
The Attempt at a Solution
I tried making 2 equations to solve them simultaneously but the values I'm looking to find always cancel out.
I know that s...
Suvat -- Two cars approach each other on a straight road
Homework Statement
Two cars approach each other on a straight road. Car A is moving at 16m/s and car B at 8m/s. When they are 45m apart both drivers apply their brakes. Car A slows down at a rate of 2m/s^2 while car B slows down at...
Homework Statement
In 4 seconds a car travels with constant velocity of 50m/s. After 4 seconds the car decelerates at 10m/s^2 until velocity = 0. What is the velocity at 8 and 12 seconds and what is the distance traveled after 12 seconds?
Also how would I draw the acceleration time graph...
Hi, I'm just practicing some SUVAT style questions as part of my revision and have a quick question.
Here's an example, a ball is catapulted vertically upwards at 10m/s, find the time it takes to hit the ground.(Discount air resistance)
I used S =0 U=10 m/s V=? A=-9.8 T=?
Using v =u +at...
A fairground ride ends with the car moving up a ramp at a slope of 30 degrees. Given that the deceleration of the car is 4.905 m/s, and that the car enters the bottom of the ramp at 18 m/s, calculate the minimum length of the ramp for the car to stop before it reaches the end.
It seems if I...
Homework Statement
Calculate the time taken for particle A to catch up with particle B.
See attachment for variables.
I worked out the answer by finding the relative velocity of A wrt B but apparently there's another method I should have/could have used to solve it?
Homework Equations...
Homework Statement
Gravity is assumed to be 10ms-2 throughout.
A ball is thrown vertically upward with a speed of 14ms-1. Two seconds later, a ball is dropped from the same point find where the two balls meet.
(They will meet when they are the same displacement (s) from the starting...
Homework Statement
A cricket ball is thrown vertically upwards and returns in 4.2s. Find (a)The initial velocity of the ball
(b)The maximum height reached
Homework Equations
v=u+at
v^2=u^2+2as
s=(u+v)/2 multiplied by t
s=ut+1/2at^2
The Attempt at a Solution
s=?
u=?
v=...
A Ball is thrown vertically upwards at 20 m/s, find its displacement after:
a) 1 second
b) 5 seconds
c) What is the maximum height the ball can reach
(Take the acceleration due to gravity to be of 10 m/s)
a) I did: s = ut + 0.5 x at^2
I got 15 m, which I'm pretty sure is correct.
b)...
Homework Statement
Diver enters the water at 7.92 [ms][/-1] and comes to rest 1.6m below the surface.
Calculate the average total upward force acting on the diver which brings his verticle velocity to zero.
From an earlier part of the questions the divers mass is 72kg.
Homework...
Homework Statement
A stone is projected horizontally from the top of a vertical sea cliff 49m high, with a speed of 20ms^-1. Neglecting air resistance, calculate:
The time that it takes to reach the sea.
Homework Equations
v² =u²+2as
t=v-u/a
t=d/s
The Attempt at a Solution...
Hey guys,
I'm taking A Level Physics, at the moment in AS level (lower level) studying kinematics, pretty straight forward but I am having trouble with projectile motion.
I can't think of a question but this is like the question I had on my physics paper the other day.
A ball is thrown...
Homework Statement
A mass accelerates from 2.5ms^(-1) to 7.5ms^(-1) over a distance of 45m. Find the acceleration and the time it takes to do this.
Homework Equations
I know I need to use s=0.5(u+v)t
But I have no idea how to rearrange it to find t, I am useless at rearranging...
When quoting the suvat equations for vectors, are they exactly the same as the normal ones, apart from the fact that the letters (apart from t) have "lines" underneath them?
Thanks
okay, this is simple and i feel ridiculous for asking, but I'm reviewing for my final and i do not remember how to do this type of problem..
Runner A runs toward a flag pole at a constant speed of 5.0 m/s. Runner B begins to run toward the flagpole at 3.0 m/s^2 when Runner A is 15 m ahead...
Hi all,
I am new here with probably what is a very simple question for most. I hope I am posting this in the right place. The question is:
A ball is thrown vertically into the air, and leaves the thrower's hand when it is 1.6m above the ground. It hits the ground 3.1s later.
Assume...
Hey, I'm currently studying for an AS exam in Physics under the OCR examination board, i got an exam in like 10 weeks and I am doing ok with physics, the only place i need to develop my understanding is using SUVAT equations, i understand them, memorized them but i haven't practised them...
I am currently doing a project involving an object sliding down a slope and falling freely for a given height afterwards. The horizontal distance is recorded, as is the angle of the slope. I have been trying to create an equation for u (the initial speed) from this data, but I haven't been able...