High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Physics Textbook (Master) > Motion in 1/2/3D: Kinematics

Basic Concepts of Kinematics

Things to know about Vectors - Building & ManipulationPrevious Section
Motion DiagramsNext Section
Popular Courses
Find My Course
0:00 / 0:00

Distance and Displacement


Kinematics is the study of motion and changes in motion. Displacement and distance are variables which are used to show the change in position, so they are talking about length.

  • Position of an object is the location of the object in space and is usually shown by a vector starting from the origin and ending at the location of the object. This vector is usually shown by x⃗\vec{x}
  • Displacement of an object between two points is a vector which connects the initial point to the final point and is usually shown by Δx⃗\Delta\vec{x}
  • Displacement is a vector and has magnitude and direction
  • Direction of displacement is from the initial point to the final point
  • The overall length of the path travelled by a particle is distance and it is usually shown by dd

Wize Tip
Distance is a scalar value and is path-dependent. Displacement is a vector value and gives the change in position between two time intervals so it is path-independent.

  • Displacement between two positions x1⃗\vec{x_1}and ​​x2⃗\vec{x_2} could be obtained using the following equation:

Δx12→=x2→−x1→\Delta\overrightarrow{x_{12}}=\overrightarrow{x_2}-\overrightarrow{x_1}













PAGE BREAK

Total Displacements and Total Distances

Given a system with ii steps,
  • Final Displacement Δx⃗=∑Δx⃗i\Delta\vec{x}=\sum_{ }^{ }\Delta\vec{x}_i
  • Total Distance d=∑∣Δx⃗i∣d=\sum_{ }^{ }\left|\Delta\vec{x}_i\right|

0:00 / 0:00

Speed and Velocity


Speed and Velocity are used to describe a change in distance and displacement with respect to change in time. The SI unit used for these quantities is [ms]\left[\frac{m}{s}\right] .

  • Average speed is a scalar quantity and defined as the overall distance traveled in time interval of Δt\Delta tdivided by this time. Speed is always positive.

speed=dΔt[ms]\boxed{speed=\frac{d}{\Delta t}\left[\frac{m}{s}\right]}

  • dd is the distance travelled in meters

  • Average velocity is a vector and defined as the ratio of displacement between two points and the time interval between two points:


vˉ=Δx⃗Δt=xf→−x0→tf−t0\boxed{\bar{v}=\frac{\Delta\vec{x}}{\Delta t}=\frac{\overrightarrow{x_f}-\overrightarrow{x_0}}{t_f-t_0}}
  • The direction of the average velocity is the same as the direction of displacement vector and is called the direction of motion!
  • Average velocity is also called the rate of change in position
  • Velocity can change in both direction and length (linear motion vs circular motion!)




  • Velocity and speed are often given in km/hkm/h or mphmph and should be converted into m/sm/s. This relationship might be useful for you.
3.6kmhr=1[ms]3.6\frac{km}{hr}=1\left[\frac{m}{s}\right]

Exam Tip
For approximation purposes it's useful to remember that 100km/h100km/h is ∼30m/s\sim 30m/s


Wize Tip
Instantaneous speed is still defined as distance divided by time, but the time interval being considered is as short as possible (or "infinitesimal") and it talks about speed at the moment!



0:00 / 0:00

Acceleration


  • Acceleration is defined as the rate of change in velocity:

aˉ=Δv⃗Δt=(vf→−v0→)tf−t0\boxed{\bar{a}=\frac{\Delta\vec{v}}{\Delta t}=\frac{(\overrightarrow{v_f}-\overrightarrow{v_0})}{t_f-t_0}}
  • Direction of the average acceleration is the same as the direction of Δv⃗\Delta\vec{v}

Watch Out!
Acceleration, velocity, and displacement do not need to point in the same direction!

  • An object is speeding up if and only if acceleration and velocity are pointing in the same direction.




  • An object is slowing down if and only if acceleration and velocity are pointing in opposite directions.




  • An object is moving at a constant velocity if and only if a=0a=0





PAGE BREAK

Motion in One Dimension

Motion in one dimension means that the particle can only go backward and forward
  • For one-dimensional problems vectors could be shown by positive and negative numbers representing pointing forward and backward respectively
  • The origin is an arbitrary place with x=0x=0
  • By convention, x>0x>0for points to the right of or above the origin and x<0x<0for points to the left of or below the origin
  • aa and vv have positive sign if they are pointing toward positive xx
  • aa and vv have negative sign if they are pointing toward negative xx

0:00 / 0:00

Example: Acceleration


A car is travelling at 20 m/s and the driver applied the brake for 10 seconds before the car comes to a stop.

a) What is the acceleration (deceleration) of the car?
b) While the car is slowing down, in what directions are the velocity and acceleration vectors pointing?

It's always nice to write down what we already know.

vo=20 m/sv_o=20\text{ m/s}
vf=0 m/sv_f=0\text{ m/s}
a=?a= ?
t=10 st=10\text{ s}

Part a)

Now to get the acceleration we just need to remember it is the change in velocity, over change in time.

a=Δv/Δta=\Delta v/\Delta t
a=(0)−(20m/s)10sa=\frac{(0)-(20 m/s)}{10 s}
a=(−20m/s)/10s=−2m/s2a=(-20\text{m/s})/10\text{s}=-2\text{m/s}^2

So the acceleration of the car is -2m/s2 , the negative is telling us that it is opposite our velocity direction. This makes sense! We are decreasing our velocity as we brake.

Part b)


The car is still moving forward while it is slowing down. That is, the velocity points forward while acceleration points backward.



James is off on a rocket. He starts off at Earth, and launches off at 9am. He flies up 200km by 10am, and then another 150km by noon. He returns to Earth at 1:30pm. Give all answers in SI units.


a)What was his average speed for the whole trip?
b) What was his average velocity for the whole trip?
c) What was his average velocity from 10am to 1:30pm?
Extra Practice
If you walked 3.00km West, and then 5.00km South to get to the store, what angle are you traveling at relative to the South direction? What is the magnitude of your displacement?
If you walked 3.00km West, and then 5.00km South to get to the store, what angle are you traveling at relative to the South direction? What is the magnitude of your displacement?
Practice question: position, velocity, and acceleration
The position of a particle is given by
r⃗(t)=(10t+t2+7)m\vec{r}(t)=(10t+t^2+7)m
a) Calculate instantaneous velocity at t =5 sec
Practice question: position, velocity, and acceleration
The position of a particle is given by
r⃗(t)=10t+t2+7\vec{r}(t)=10t+t^2+7
Calculate instantaneous velocity at t =5 sec, average velocity for interval (3-5 sec). Also calculate instantaneous acceleration at t =1 sec.
Forces: Friction
A professional mover drags a 20kg mattress across the floor. The mover exerts a force of 50 N at 30° to the horizontal. The coefficient of kinetic friction is 0.100. What is the acceleration of the mattress?
I'm moving and packed all my books into a box that weighs 17kg. I pull the box on a weightless dolly with a force of 60N at an angle of 30° to the horizontal surface.
a) What is the acceleration of the box?
b) The box feels the normal force from the ground and the force of gravity. Are the normal force and gravity an action-reaction pair? Explain your answer!
Practice: Distance and Displacement
You drive 200 kilometres west to get from Montreal to Ottawa. On the way back to Montreal, you run out of gas after driving 50 kilometres.
Two runners (Bob and Alice) are racing. Alice is much faster than Bob and is placed such that she starts 25 m25\ mbehind Bob. She is 1.51.5times faster than Bob who runs at 8 m/s8\ m/s.
a) If they both start at the same time, when are these runners at the same point on the track?
b) Who will win in a 1000 m1000\ mrace if Alice has to start 200 m200\ mbehind? Explain your answer.
Throwing a rock down a well
Joe throws a rock straight down a well. He throws it at 13.0m/s, and the well is 5.10m deep. What is the velocity of the rock when it gets to the bottom? Use the conventional coordinates, where up is positive.
Franklin is coming over to your house. He first travels at 3m/s East, and then he travels 3m/s South. What direction is his average acceleration?
In horse racing, units of "horses" are used to measure length. It is defined as the length of a horse, or 8ft. And there are 3.28ft in 1m.
In the 1973 Belmont race, the horse Secretariat (great movie) won by 31 horse lengths. What is that distance in cm?
Riding a ferris wheel
Joanne is riding a ferris wheel that is 100.0m in diametre. Starting at the bottom, she rides 2 and 3/4 revolutions before the wheel gets stuck.
If you walked 3.00km West, and then 5.00km South to get to the store, what angle are you traveling at relative to the South direction? What is the magnitude of your displacement?
A rabbit, escaping a fox, runs 40o north of west at 10.0m/s. A coordinate system is established with the positive x-axis to the east and the positive y-axis to the north. Write the rabbit’s velocity in terms of components and unit vectors.
Taron is off on a rocket. He starts off at Earth, and launches off at 9am. He flies up 200km by 10am, and then another 150km by noon. He returns to Earth at 1:30pm. Give all answers in SI units.
(NB: the video says the trip ends at 2:30pm, this is incorrect. Should be 1:30pm, and the mathematics is still fine.)
a) What was his average speed for the whole trip?
Vanessa is making her way downtown, moving quite fast. She's racing Adam who's in a big yellow taxi. Their velocities are shown below
Over the entire race, who travelled the longest distance? Who has the greater displacement?
Average acceleration of runner
Frank is running at 3.0m/s. He accelerates until he is travelling at -1.3m/s. What is his average acceleration if it took him 12.0s to do this?
A bird flew 100m to the east then 200m to the northwest. Use the algebraic addition of vectors to find the bird’s net displacement.
A 20.0kg cart is moving at 10.0m/s before it goes down a hill. It ends up 2.0m below where it started, and travelled a distance of 15.0m. What is the speed of the car? There is friction, and it applies an average force of 2.00N over the course of it's motion.
Give your answer in m/s.
I over-packed a box and it weighs 50kg. I'm pulling it across a rough surface at an angle of 45 degrees above the horizontal with a force of 280N. The coefficient of kinetic friction of the floor is 0.5. I pull for 10 seconds, the box speeds up from rest, I release it and it continues to slide to a halt. What is its maximum speed?
Area under plot for displacement
Alice and Bob are having a race. Alice starts off strong, at 10.0m/s, while Bob struggles at 5.0m/s. But 5 minutes later, Alice starts to tire out, and slows to 2.0m/s. Bob sees his chance, and starts running at 7.0m/s. The race ends another 5 minutes later. Who travelled the furthest, winning the race?
Direction of acceleration
Joe is running from 20m to 10m, while he is slowing down. What direction is his acceleration?
If you walk around a circle of diameter of 10 m for π rad, how far have you walked?
Enter answer in meters.
Kinematics question
The slope of position-time graph gives
average velocity
Practice question: position, velocity, and acceleration
The position of a particle is given by
r⃗(t)=10t+t2+7\vec{r}(t)=10t+t^2+7
Calculate instantaneous velocity at t =5 sec, average velocity for interval (3-5 sec). Also calculate instantaneous acceleration at t =1 sec.
A 200g mass block is released from rest at a height of 25 cm on 30o incline. The incline is frictionless. It slides down the incline and then a frictionless surface until it collides elastically with an 800g block at rest 1.4 m from the bottom of the incline as shown in the figure. How much later do the two blocks collide again?
If the velocity of particle is non-zero. Can acceleration of particle be zero?
If the average velocity of an object is zero in some time interval, object's displacement is
zero
The position of a particle of mass 2g is given by r⃗=−3t2i^+2tj^+5k^\vec{r} = -3t^2\hat{i}+2t\hat{j}+5\hat{k} (t is in sec and r is in m).
The magnitude of instantaneous velocity and the magnitude of instantaneous acceleration at t = 3s are:
Practice question: position, velocity, and acceleration
The position of a particle is given by r⃗(t)=(10t+t2+7)m\vec{r}(t)=(10t+t^2+7)m
a) Calculate instantaneous velocity at t =5 sec
b) Calculate the average velocity for an interval (3-5 sec).