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Wize University Physics Textbook (Master) > Motion in 1/2/3D: Kinematics

One-Dimensional Kinematic Equations

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One-Dimensional Kinematics


Velocity, displacement, and acceleration can be related using the kinematic equations of motion. These equations are different depending on type of the motion in the problem. Here, we introduce kinematic equations of motions with constant velocity and constant accelerations in one dimension.


Motion with a Constant Velocity

  • For constant velocity motions (a=0a=0):
x−x0=vt\boxed{x-x_0=vt}

Motion with a Constant Acceleration

  • For constant acceleration motions, there are three useful formulas:
x−x0=v0t+12at2\boxed{x-x_0=v_0t+\frac{1}{2}at^2}

v=v0+at\boxed{v=v_0+at}

vf2=v02+2a(x−x0)\boxed{v_f^2=v_0^2+2a(x-x_0)}


  • When solving a problem, we can use any of the above equations depending on the information given and asked for in the problem.

Wize Tip
  • Kinematic equation can be used when acceleration is constant i.e a does not change with particle motion.
  • In order to find the right equation to use, look for the missing variable.


Watch Out!
It is crucial to pick an appropriate sign for each of the variables in above equations since they are all vectors!



Steps to Take to Solve Any Kinematics Problem:

  1. Draw and label a diagram.
  2. Define the coordinate system (doesn't matter what you choose, as long as you are consistent).
  3. Identify the known values with units.
  4. Pick the right equation. Notice how if you know any 3 of Δx⃗,v⃗o,v⃗f,a⃗,t\Delta \vec{x}, \vec{v}_o, \vec{v}_f, \vec{a}, t , you can entirely solve the question!
  5. Solve for the quantity you want. Write down all your steps!
  6. Think about the final answer you got. Do the signs and value make physical sense?


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Example: Stopping Distance of a Driving Car


A car can decelerate at 4.0 m/s2m/s^2. If somebody is driving this car at 110 km/h and human reaction time is 0.45 seconds, what will the stopping distance be for the driver?


Solution:

First, let's convert the speed: 110 km/h = 30.6 m/s.

There are two parts to the motion: first, we continue for 0.45 seconds at the current speed. Then, we continue to move forward while decelerating.

First part: constant speed, time is known, we want distance.
x=vt=(30.6m/s)(0.45s)=13.8mx=vt=(30.6 m/s)(0.45 s)=13.8 m
Second part: the acceleration is in the opposite direction of the velocity, so it needs to be negative. We know the final speed, initial speed, and acceleration, but not time. We want distance.

vf2=vo2+2axx=vf2−vo22ax=(0)2−(30.6m/s)22(−4.0m/s2)x=117mv_f^2=v_o^2+2ax \newline x=\frac{v_f^2-v_o^2}{2a} \newline x=\frac{(0)^2-(30.6 m/s)^2}{2(-4.0 m/s^2)} \newline x=117 m
We add these two distance together now: xtotal=13.8m+117m=131mx_{total}=13.8m+117m=131m


A vehicle travelling 130 km/h passes a police car at rest. As soon as the vehicle passes the police car, the police car turns its lights on and begins to accelerate to its maximum speed of 140 km/h.

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If the police car accelerates at a constant acceleration for 12 seconds, how far will it have traveled before reaching its maximum speed?
Extra Practice
Dave drives a truck at a constant 20 m/s and passes me on my stationary motorbike, at which point I lay on the gas and accelerate at 8.0 m/s2. How far has my motorbike driven when I pass Dave?
Practice: Drone flight
A drone was controlled as follows:
I. At t=0 st=0\ s, the drone was stationary on the surface of the ground (y=0y=0)
II. At t=5 st=5\ s, it was raised up at a constant speed of 4 m/s4\ m/s
Motion in 1D and 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.
Kinematics
Your dog, Buttercup, sees your neighbour's cat (Waffles) running at its maximum speed of 4 m/s4\ m/s. If your dog can go from 0 to 60 km/h in a time of 15 seconds, will Buttercup catch Waffles before the 35m35mdriveway ends? Show your work.
Kinematics
A bike speeds up from 10.0km/hr10.0 km/hrto 32.0km/hr32.0 km/hrin 15.0s15.0 s. The acceleration is constant.
a) What is the acceleration of the bike? Enter your answer in m/s2.
b) How much distance was traveled in that time? Enter your answer in meters.
Kinematics: Braking Distance
A car can decelerate at 4.0 m/s2m/s^2. If somebody is driving this car at 110 km/h and human reaction time is 0.45 seconds, what will the stopping distance be for the driver?
Using the equations and problem-solving steps: Police chase
Let's illustrate the use of kinematic equations and problem-solving steps with an example.
A vehicle travelling 130 km/h passes a police car at rest. As soon as the vehicle passes the police car, the police car turns its lights on and begins to accelerate to its maximum speed of 140 km/h.
A drone was controlled as follows:
I. At t=0 st=0\ s, the drone was stationary on the surface of the ground (y=0y=0)
II. At t=5 st=5\ s, it was raised up at a constant speed of 4 m/s4\ m/s
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 block is sliding on gym floor. The coefficient of static and kinetic friction between the block and the floor are 0.40.4and 0.070.07 respectively. If the block is initially traveling at 3 m/s3\ m/s find the total distance the block must travel to stop. Draw and label a FBD for both the initial state and the final state.
If a bullet leaves the muzzle of a rifle at 600.0 m/s, and the barrel is 0.90 m long, what was the acceleration of the bullet while in the barrel?
Enter your answer in scientific notation in the following form: 1.0x10^2. Don't include units in your answer (units are in m/s2)
Practice question: kinematic equations
A bullet is fired through a board 10.0 cm thick in such a way that the bullet’s line of motion is perpendicular to face of the board. If the initial speed of the bullet is 400 m/s and it emerges from the other side of the board with a speed of 300 m/s, find (a) the acceleration of the bullet as it passes through the board and (b) the total time the bullet is in contact with the board.
A ball is thrown with an initial upwards velocity of v0v_0from a height hh(both of these values are positive).
a) What is the maximum velocity during the motion?
b) What is the point of minimum acceleration?
Practice: Acceleration
A drone was controlled as follows:
I. At t=0 st=0\ s, the drone was stationary on the surface of the ground (y=0y=0)
II. At t=5 st=5\ s, it started to raise up at a constant speed of 4 m/s4\ m/s
Your dog Buttercup sees your neighbour's cat (Mr. Waffles) running past at its top speed of 4 m/s4\ m/s. If your dog has an impressive 0−60 km/h0-60\ km/hof 15s15 swill Buttercup catch Mr. Waffles before the 35m35mdriveway ends?
Two runners (Bob and Alice) are racing. Alice is much faster than Bob and is placed such that she starts 25 m25\ m behind Bob. She is 1.51.5 times 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\ m race if Alice has to start 200 m200\ m behind?
A bike accelerates constantly from 10km/hr10km/hr to 32km/hr32km/hr in 15s15 s. What is the acceleration of the bike? How much distance was traveled in that time?
Dave drives a truck at a constant 20 m/s and passes me on my stationary motorbike, at which point I lay on the gas and accelerate at 8.0 m/s2. How far has my motorbike driven when I pass Dave?
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? How long until it stops? What is the box's total displacement?
Kinematic equations
If a bullet leaves the muzzle of a rifle at 600.0 m/s, and the barrel is 0.90 m long, what was the acceleration of the bullet while in the barrel?
Enter answer in m/s2
Practice question: kinematic equations
A bullet is fired through a board 10.0 cm thick in such a way that the bullet’s line of motion is perpendicular to face of the board. If the initial speed of the bullet is 400 m/s and it emerges from the other side of the board with a speed of 300 m/s, find:
(a) the acceleration (in km/s2 ) of the bullet as it passes through the board and
(b) the total time (in sec) the bullet is in contact with the board.
A speeding car is traveling at a constant 50 m/s when it passes a stationary police car. If the police car delays 2s before starting, what must be the magnitude of the constant acceleration of the police car to catch the speeding car after the police car travels a distance of 500 m?
A car is travelling at 20m/s and the driver applied the brake for 10 seconds before the car comes to a stop. What is the net acceleration (deceleration) of the car?