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B. Kinematics · Kinematics: Motion in One and Two Dimensions

Kinematics: Motion in One and Two Dimensions: Unit Review

A1.1 A1.2 A1.3 A1.4 A1.5 A1.6 A1.7 A1.8 A1.9 A1.10 A1.11 A1.12 A1.13 A2.1 A2.2 B1.1 B1.2 B2.1 B2.2 B2.3 B2.4 B2.5 B2.6 B2.7 B2.8 B2.9 B3.1 B3.2 B3.3About 40 minutes
Learning goals
  • I can describe and graph one-dimensional motion and move between d–t, v–t and a–t graphs.
  • I can derive and use the equations of uniformly accelerated motion, including free fall.
  • I can solve two-dimensional displacement, velocity and projectile problems with vectors.
  • I can plan and analyse a motion inquiry and assess a kinematics technology.

Words to know

displacement
Change in position, Δd = d₂ − d₁ (a vector).
velocity
Rate of change of position, vₐᵥ = Δd ÷ Δt (a vector).
acceleration
Rate of change of velocity, aₐᵥ = Δv ÷ Δt (a vector).
uniformly accelerated motion
Motion with constant acceleration; a straight v–t graph.
projectile
An object moving under gravity alone; constant horizontal velocity and constant vertical acceleration.

Let's review!

In this unit you learned the language of motion: scalars and vectors, position, displacement, velocity and acceleration. You read and built position-time, velocity-time and acceleration-time graphs, linked by slopes and areas, and used them to derive the five equations of uniformly accelerated motion. You solved one-dimensional problems, including free fall and stopping distances, and planned and analysed your own motion inquiry. Then you moved into two dimensions with vector diagrams and components, river and wind problems, and projectiles launched horizontally and at an angle. Finally you looked at the technologies that measure motion and how they affect society.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Work through one or two sections at a time. For every calculation, write the Given, Required, Analysis, Solution, Statement steps and state your positive direction. Draw a sketch or graph for every question, even when one is not asked for. Keep your calculator in degree mode and use g = 9.8 m/s². Round only final answers, to sensible significant digits. For each question you miss, write down why (wrong equation? sign? units? component?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Graphsslope of d–t = vslope of v–t = aarea under v–t = Δdarea under a–t = ΔvConstant accelerationv₂ = v₁ + aΔtΔd = ½(v₁ + v₂)ΔtΔd = v₁Δt + ½aΔt²v₂² = v₁² + 2aΔdΔd = v₂Δt − ½aΔt²Vectors in 2-D[N 30° E]: face N, turn 30° to Ecomponents: Ax = A cos θ, Ay = A sin θR = √(Rx² + Ry²)Δv = v₂ + (−v₁)Projectileshorizontal: ax = 0, Δdx = vxΔtvertical: a = 9.8 m/s² [down]time links the two motionslevel ground: 45° gives max range
Your kinematics toolbox. Graphs and equations describe one-dimensional motion; vectors and components extend it to two dimensions; projectiles combine both.
Just for fun

You now know how fast a ball is moving at the top of a throw, how long a car needs to stop on snow, and how to cross a river without a map. You also know that the correct answer to "How far did you run?" is "Do you mean distance or displacement?" (Use this one carefully.)

Describing Motion: Scalars, Vectors, Position and Displacement

1

A bus goes 4.0 km [E], then 6.0 km [W]. Find its distance and displacement.

2

Is temperature a scalar or a vector? Is velocity?

3

Round 0.04050 m × 3.2 to the correct significant digits.

Speed and Velocity: Average, Constant and Instantaneous

1

A runner does 2 laps of a 400 m track in 3.0 min, finishing at the start. Find average speed and average velocity.

2

Convert 108 km/h to m/s.

3

A car moves at a constant 25 m/s [S] for 8.0 s. Find its displacement.

Position-Time Graphs: Slope, Uniform Motion and Instantaneous Velocity

1

A straight d–t line runs from (0 s, 30 m) to (6.0 s, 6.0 m). Find the velocity (east positive).

2

On a curved d–t graph, the tangent at t = 2.0 s passes through (0 s, −2.0 m) and (3.0 s, 10.0 m). Find the instantaneous velocity at 2.0 s.

3

Two lines on a d–t graph cross. What does that mean?

Acceleration and Velocity-Time Graphs

1

A puck slows from 9.0 m/s [E] to 3.0 m/s [E] in 2.0 s. Find its acceleration.

2

A v–t graph goes from (0, 2.0 m/s) to (4.0 s, 10.0 m/s) in a straight line. Find a and Δd.

3

v = +5 m/s and a = −2 m/s². Speeding up or slowing down?

Moving Between Graphs: Position-Time, Velocity-Time and Acceleration-Time

1

A v–t graph is a horizontal line at 4 m/s. Describe the a–t and d–t graphs.

2

An a–t graph shows 2.0 m/s² for 3.0 s. The object started at 1.0 m/s. Find v₂.

3

A d–t graph is a curve that is getting less steep. What is happening to the velocity?

Deriving the Equations of Uniformly Accelerated Motion

1

Which feature of a v–t graph gives v₂ = v₁ + aΔt?

2

Which equation leaves out Δt?

3

Explain where the ½aΔt² term in Δd = v₁Δt + ½aΔt² comes from.

Solving Problems with Uniformly Accelerated Motion

1

A car accelerates from rest at 2.0 m/s² over 100 m. Find its final speed.

2

A train at 30 m/s brakes at 1.5 m/s². How long does it take to stop, and how far does it go?

3

A driver at 25 m/s has a 0.80 s reaction time and brakes at 5.0 m/s². Find the stopping distance.

d–t graphv–t grapha–t graphslopeareaslopeareaMoving between motion graphsGoing right: find slopes. Going left: find areas (and add the starting value).
The three motion graphs are linked: take slopes to move from d–t to v–t to a–t, and areas to move back.

Motion in a Vertical Line: Free Fall and Objects Thrown Up

1

A ball is dropped from 44.1 m. How long does it fall?

2

A ball is thrown up at 24.5 m/s. Find its time to the top and maximum height.

3

What is the acceleration of a ball at the top of its flight?

Investigating Motion: Planning, Measuring and Analysing a Kinematics Inquiry

1

A d versus t² graph for a cart released from rest has slope 0.35 m/s². Find a.

2

Your measured g is 9.62 m/s². Find the percent error.

3

Ticker intervals (0.10 s each) are 3.0, 3.0, 3.0 cm. Describe the motion.

Vectors in Two Dimensions: Directions and Scale Diagrams

1

Write [S 70° E] another way.

2

Add 8.0 m [N] and 15 m [E].

3

What is the opposite of 12 m/s [W 30° N]?

Vector Components and Two-Dimensional Motion Problems

1

Find the components of 50 m [S 30° E] (east and north positive).

2

A boat aims straight across a 150 m river at 3.0 m/s; the current is 1.2 m/s. How long to cross, and how far downstream?

3

A car goes from 10 m/s [N] to 10 m/s [W] in 2.0 s. Find aₐᵥ.

Projectile Motion: Objects Launched Horizontally

1

A ball rolls off a 1.8 m ledge at 3.0 m/s. Find its time in the air and its range.

2

What is the ball's horizontal velocity just before it lands?

3

Two balls leave the same table at different horizontal speeds. Which lands first?

Projectiles Launched at an Angle: Airtime, Maximum Height and Range

1

A ball is launched at 20 m/s at 60° on level ground. Find its airtime.

2

Find that ball's range.

3

Which other angle gives the same range at the same speed?

Kinematics Technology and Society: Speedometers, Radar, GPS and Speed Control

1

Skid marks are 32 m long; the car slowed at 6.25 m/s². Find the starting speed.

2

Why does a radar gun need the Doppler effect but a lidar gun does not?

3

Give one benefit and one concern of truck speed limiters.

Check your understanding

Quiz

1.

Which pair contains a scalar and a vector, in that order?

2.

The slope of a velocity-time graph gives:

3.

A v–t graph is a straight line from (0, 0) to (6.0 s, 12 m/s). The displacement is:

4.

A cyclist accelerates from 4.0 m/s to 10.0 m/s in 3.0 s. Her acceleration is:

5.

A car at 20 m/s brakes to a stop at 5.0 m/s². Its braking distance is:

6.

A ball is thrown straight up. At its highest point:

7.

A hiker walks 6.0 km [N] and then 8.0 km [E]. Her displacement is:

8.

A boat aims straight across a 90 m river at 3.0 m/s. A current of 2.0 m/s flows downstream. The crossing time is:

9.

A ball is launched horizontally at 8.0 m/s from 4.9 m above level ground. Its range is:

10.

On level ground with no air resistance, which launch angle gives the same range as 35°?

For parents and teachers

Use the review to find what to revisit:

  • Mixing up distance and displacement, or scalars and vectors; significant-digit errors → Lesson 1 (Describing Motion).
  • Confusing average speed and average velocity, or unit conversions → Lesson 2 (Speed and Velocity).
  • Misreading d–t graphs, tangents and secants → Lesson 3 (Position-Time Graphs).
  • Sign errors with acceleration; slopes and areas of v–t graphs → Lesson 4 (Acceleration and Velocity-Time Graphs).
  • Sketching one graph from another → Lesson 5 (Moving Between Graphs).
  • Not knowing where an equation comes from → Lesson 6 (Deriving the Equations).
  • Choosing the wrong equation, missing "from rest" or "stops", stopping-distance stages → Lesson 7 (Solving Problems).
  • Free-fall sign errors, or saying a = 0 at the top → Lesson 8 (Free Fall).
  • Variables, linearization, percent error and sources of error → Lesson 9 (Investigating Motion).
  • Writing directions and drawing scale diagrams → Lesson 10 (Vectors in Two Dimensions).
  • Sine/cosine mix-ups, missing negative signs, river and wind problems → Lesson 11 (Vector Components).
  • Using the launch speed in the wrong direction for horizontal launches → Lesson 12 (Objects Launched Horizontally).
  • Airtime, maximum height and range at an angle; uneven launch and landing heights → Lesson 13 (Projectiles Launched at an Angle).
  • How speed technologies work and weighing their impacts → Lesson 14 (Technology and Society).

Next, the Forces unit explains why objects accelerate (Newton's laws), and the Work, Energy and Power unit looks at motion through energy. Grade 12 Physics (SPH4U) extends kinematics to circular motion and more complex two-dimensional problems.