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Linear Kinematics

1 and 2 dimensional kinematics. Learn the chart method to solve the hardest projectile problems.

Here’s everything you need to know to master kinematics:

Concept Overview:

  1. Key variables: displacement, distance, velocity, speed, acceleration, and time.
  2. Kinematic Equations
  3. Average speed vs average velocity
  4. Motion in one dimension – cars, balls, and any object moving in one direction
  5. Motion in two dimensions – object dropped from a moving plane, a bullet fired horizontally, balled kicked at 45°, and all types of projectile motion.
  6. Graphs: displacement vs time, velocity vs time, acceleration vs time.

(1) Key Variables:

Displacement (a vector quantity) is how far an object has moved from its origin. Distance (a scalar quantity) is the total amount traveled.

Suppose you took a 200-mile trip to LA and back. Your displacement is zero (because you’re back where you started). Your distance, however, is 400 miles (200 miles to and 200 miles back)

Velocity (a vector quantity) is how fast displacement changes. Speed is how fast distance changes. For example: 10 meters every second is written as 10 m/s.

Acceleration (a vector quantity) is how fast velocity changes. For example: Slowing down 5 m/s every second, written as 5 m/s2.

Time is often overlooked, but it’s the key to solving problems that involve motion in two dimensions. Why? Because time is the SAME for both the X and Y direction.

(2) Kinematic Equations:

Couple of things to remember regarding variable notation:

  1. = delta. It means “change in,” which literally means “final – initial.”
  2. s = distance. ∆x = change in horizontal displacement. ∆Y = change in vertical displacement
  3. vo = initial velocity. It is pronounced “V knot.”
  4. Equations apply both horizontally and vertically.
  5. Keep track of the directions within the equation, by using subscripts. For example, initial velocity in the vertical direction can be written as, “voy“.

The 5 fundamental Kinematic Equations

Memorize these so you won’t waste any time using the formula sheet on tests! The best way to memorize is to write them down on a flashcard and practice 20 kinematic problems.

(3) Average Speed vs Average Velocity

This topic seems to confuse students a bit. There is big difference between average speed and velocity.

Average speed is distance divided by time. Average velocity is displacement divided by time.

If you run 100 meters around a track in 5 minutes:

a) your average speed is 100m/5mins = 20m/min.

b) your average velocity is 0m/5mins = 0m/min. (Becuase displacement is 0!)

(4) Motion in 1 dimension

Any object moving in a straight line ( up, down, left, right ) is moving in one direction. A car that starts from rest and accelerates to 80 m/s in 30 seconds, is moving in one direction.

Let’s say we are asked to find the distance the car traveled in 30 seconds. Start by choosing a kinematic equation that has all the given variables, then plug the numbers in. It’s as simple as that!

(4.1) Picking the right equation

Here’s a little trick if you can’t figure out what equation to use:

There are no more than four variables in each kinematic equation. Similarly, every kinematics problem has four variables: three known and one unknown variable)

Write out the 4 variables and see which kinematic equation has all 4. In the example above we have: vo , vf , t , and ∆x. Thus you should use the equation: ∆x = (vf – vo)t.

(5) Motion in 2-dimensions

Objects moving in 2-dimensions, are moving both horizontally and vertically. These objects are called projectiles. Solve problems the exact same way as with one-dimensional kinematics, except apply the formulas in both directions.

(5.1) A few helpful tips to remember:

  1. Time stays the same in both the x and y directions
  2. Horizontal speed, distance, or time, can be found using ∆x = vt
  3. The vertical acceleration is almost always, acceleration due to gravity (9.81 m/s)
  4. Unless given, the initial velocity is almost always 0.
  5. Vertical velocity at the top of the trajectory is always 0, but horizontal velocity remains constant.
  6. If a projectile is traveling at an initial angle, use trig to split the vector into horizontal and vertical components.
  7. The final velocity of the object is the SUM of the horizontal and vertical components of velocity

(5.2) Example of 2d Kinematics

A bullet shot horizontally at 90 m/s, from a height of 3.2 meters. Find how far the bullet travels.

To solve this problem just figure out what’s given in the horizontal and vertical directions, and pick a kinematic equation to solve for the unknown. Here’s an example of what the work might look like:

Horizontal ComponentsVertical Components
vox = 90 m/s voy = 0 m/s
t =?ay = 9.81 m/s2
∆x =?∆y = 3.2 m
t =?

Hopefully, you can see the pattern here. We aren’t given enough information in the horizontal direction. Therefore, we will use a kinematic equation in the vertical direction to find the time.

Remember that time is the common factor in both directions. If we solve for t, we will be able to solve for the distance the bullet travels with the equation ∆x = vox × t.

(6) Graphs

Most students overlook graphing. It comes up quite a bit on the AP Exam. I’ll show you the best way to understand how these graphs work.

acceleration vs time graph

Try this question:

The acceleration vs time graph is shown on the left. Find the total change in velocity in the first 3 seconds.

First, let’s take a step back and go back to definitions

  • Velocity is the change in displacement per unit of time. “Change in” is another word for slope/derivative! Thus the slope of the displacement vs time graph is velocity!
  • Acceleration is the change in velocity per unit of time. Thus the slope of the velocity vs time graph is acceleration.

Look at it like this: ∆x –> ∆v –> a. To get to the next one just find the slope/derivative!

Now go backwards: a –> ∆v –> ∆x. To get to the next one just find the opposite of the derivative: the integral (a fancy way of saying area under the curve!)

Are you able to solve the problem above now? If you are, great job! Keep practicing with the problems at the end of this post!

(7) Wrap up

Congrats! That wraps up everything you need to know for kinematics for students in regular and advanced Physics classes. It’s best to apply your knowledge to see if you really are understanding these concepts. Nerd-Notes has put together some practice questions that you will find super useful!

(8) Practice!

Here’s a document of 50 advanced Kinematic problems, put together by Physics pros, and based on the actual AP Physics 1 exam. If you can solve these, you will be able to solve anything!

50 Questions to Master Kinematics!

AND if you’re having trouble and want to work with a Physics PRO, join our ELITE 1-to-1 Physics mentorship program. All documents and learning materials will be included for FREE.

(9) 10 Practice Questions to help you master Kinematics

Solve all 10 questions first. Answers are given at the end of the section.

(1) The international space station travels at 7660 m/s. Find the average velocity of the space station, if it takes 90 minutes to make one full orbit around earth.

⇨ concepts involved: average velocity, problem-solving
⇨ difficulty: easy

(2) Police officers have measured the length of a car’s tire skid marks to be 23 meters. This particular car is known to decelerate at a constant 7.5 m/s2. What was the car’s initial velocity?

⇨ concepts involved: kinematic equations, problem-solving
⇨ difficulty: easy

(3) A plane, 220 meters high, is dropping a supply crate to an island below. It is traveling with a horizontal velocity of 150 m/s. At what horizontal distance must the plane drop the supply crate for it to land on the island?

(a) 589 meters
(b) 755 meters
(c) 964 meters
(d) 1005 meters
(e) 1127 meters

⇨ concepts involved: applying kinematics equations in two dimensions, projectile motion, problem-solving
⇨ difficulty: medium

(4) A rock is thrown at an angle of 42° above the horizontal at a speed of 14 m/s. Determine how long it takes the rock to hit the ground.

(a) .92 seconds
(b) 1.8 seconds
(c) 2.4 seconds
(d) 3.3 seconds
(e) 3.9 seconds

⇨ concepts involved: projectile motion, problem-solving
⇨ difficulty: hard

(5) Conceptual problem: A golfer hits her ball in a high arcing shot. Air resistance is negligible. When the ball is at its highest point, which of the following is true?

(a) acceleration and velocity are both zero
(b) the ball has zero velocity for a split second
(c) the ball’s range is at a maximum
(d) both the x and y components of velocity are zero
(e) the x component of velocity is greater than the y component of velocity

⇨ concepts involved: projectile motion, range, vector components
⇨ difficulty: easy

(6) A soccer ball is kicked horizontally off an 85-meter high cliff, at a speed of 34 m/s. What was the ball’s final speed when it hit the ground below?

(a) 53 m/s
(b) 41 m/s
(c) 1681 m/s
(d) 776 m/s
(e) more information is needed

⇨ concepts involved: projectile motion, speed
⇨ difficulty: medium

(7) Conceptual Problem: A 100-pound rock and a 1-pound metal arrow pointed downwards, are dropped from height h. Assuming there is no air resistance, which one hits the ground first.

⇨ concepts involved: free-fall, acceleration, vertical kinematics
⇨ difficulty: medium

(8) A gun can fire a bullet to height h when fired straight up. If the same gun is pointed at an angle of 45° from the vertical, what is the new maximum height of the projectile?

(a) h/4
(b) h/(2√2)
(c) h/2
(d) h/√2
(e) h

⇨ concepts involved: free-fall, projectile motion, problem-solving
⇨ difficulty: medium

(9) Conceptual Problem: Two balls are dropped off a cliff, 3 seconds apart. The first ball dropped is twice as heavy as the second ball dropped. Air resistance is negligible. While both balls are falling, the distance between the two balls is:

(a) constant
(b) increasing
(c) decreasing
(d) dependent on the height
(e) unknown

⇨ concepts involved: free-fall, projectile motion, acceleration due to gravity
⇨ difficulty: hard

(10) A arrow is shot horizontally from a distance of 20 meters away. It lands .05 meters below the center of the target. If air resistance is negligible what was the initial speed of the arrow?

⇨ concepts involved: projectile motion, problem-solving, kinematic equations
⇨ difficulty: medium

Want more conceptual, problem-solving, or graphical problems? Here is another

50 Questions to Master Kinematics!

– written by Physics Pros
– designed to make you think out of the box
– if you can solve these, you can solve any kinematic problem
– based on real AP/ advanced Physics problems

(9.1) Answers

  1. 0 m/s
  2. 18.6 m/s
  3. (d)
  4. (b)
  5. (e)
  6. (a)
  7. It takes the same time for both.
  8. (c)
  9. (b)
  10. 200 m/s

(10) More Help

Need help understanding these questions or want to excel at Physics quickly? Book a 1-to-1 lesson with me and see the results for yourself. Claim your free trial lesson here! Over 200 AP, IB, and Honors students have joined to date. You can also join other Elite Physics programs here.

If you are a beginner, consider joining our Physics in 3 Weeks Program to get ahead and stay ahead of your classmates. See upcoming dates here.

Jason Kuma
Jason Kuma

Founder, Writer, Physic B.S, Business B.A USC, Fremont CA

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