Kinematics — Motion in 1D and 2D

Big Picture

Kinematics answers four questions about any moving object: Where is it? How fast is it going? Is its speed changing? How do we predict where it will be? The answers come from three linked quantities — position, velocity, and acceleration — each of which is the rate of change of the one before it.

ELI-10: Explain It Like I'm 10

Imagine you are riding your bike down a straight street. Kinematics is just a fancy word for describing your ride: where you started, how far you went, how fast you were pedaling, and whether you sped up or slowed down. If someone asks, "How long will it take you to reach the park?" — that is a kinematics question. You do not need to know about muscles or gears or wind. You just need to describe the motion itself.

1.1 Displacement and Velocity

Core Idea

Position tells you where an object is. Displacement tells you how far it moved from its starting point, and in what direction. Velocity tells you how fast the position is changing.

Important Definitions

• Position x (or r in 2D/3D): location relative to a chosen origin. SI unit: meter (m).
• Distance: total length of the path traveled. Always positive. SI unit: meter (m).
• Displacement Δx = x_f - x_i: the straight-line change in position, including direction. SI unit: meter (m).
• Average velocity v_avg = ΔxΔt. SI unit: meter per second (m/s).
• Speed: magnitude of velocity. Always positive. SI unit: m/s.

Critical distinction: Distance is a scalar (just a number). Displacement is a vector (magnitude and direction). If you walk around the block and return to your front door, your distance might be 400 m, but your displacement is zero.

Physics and Mathematics

v_avg = ΔxΔt = x_f - x_it_f - t_i

Where:

• v_avg = average velocity (m/s)
• x_f = final position (m)
• x_i = initial position (m)
• Δt = time interval (s)

The sign of velocity tells you the direction. Positive velocity means motion in the +x direction. Negative velocity means motion in the −x direction.

Instantaneous velocity is the velocity at a single moment — it is the slope of the tangent line on a position-versus-time graph. v = lim_{Δt → 0} ΔxΔt = dxdt

Conceptual Example

A student walks 3 m east, then 4 m west, taking 7 seconds total. Her distance traveled is 7 m. Her displacement is −1 m (1 m west). Her average speed is 7/7 = 1.0 m/s. Her average velocity is (-1)/7 ≈ -0.14 m/s.

ELI-10: Explain It Like I'm 10

You are playing in your yard. Your position is where you are standing. Displacement is the arrow pointing from where you started to where you ended up — even if you ran all over the place first. Velocity is how fast that arrow grows (or shrinks) as time passes. If you end up exactly where you started, your displacement arrow has zero length, so your average velocity is zero, no matter how much you ran around.

Common Mistakes

• Confusing distance with displacement. Distance is the odometer reading; displacement is the straight line from start to finish.
• Forgetting that velocity has a sign/direction. Negative velocity does not mean slowing down; it means moving in the negative direction.

Key Takeaway

Velocity is the rate of change of position. A position graph's slope tells you velocity. Displacement is the vector change in position; distance is the scalar path length.

1.2 Acceleration

Core Idea

Acceleration is the rate at which velocity changes. An object accelerates whenever it speeds up, slows down, or changes direction.

Important Definitions

• Average acceleration a_avg = ΔvΔt. SI unit: m/s².
• Instantaneous acceleration a = dvdt: slope of the velocity-versus-time graph.
• Free fall: motion under gravity alone, with g ≈ 9.8 m/s² downward near Earth's surface.

Physics and Mathematics

a_avg = v_f - v_iΔt

The sign of acceleration relative to velocity determines whether the object is speeding up or slowing down:

• Same sign (both positive or both negative) → speeding up.
• Opposite signs → slowing down (sometimes called deceleration, though physicists prefer "negative acceleration" only when acceleration is literally negative).

Graph Interpretation — Position, Velocity, and Acceleration

Understanding graphs is as important as manipulating equations. The three kinematic graphs are deeply connected:

Position-vs-time (x-vs-t) graph:

• Slope at any point = instantaneous velocity. A steeper slope means higher speed.
• Curvature indicates acceleration. A straight line = zero acceleration (constant velocity). A curve that bends upward = positive acceleration; a curve bending downward = negative acceleration.
• If the line is flat (zero slope), the object is at rest.
• If the line slopes upward, the object moves in the +x direction; if downward, it moves in the -x direction.

Velocity-vs-time (v-vs-t) graph:

• Slope = acceleration. A straight sloping line indicates constant acceleration.
• Area under the curve = displacement Δx. Count area above the time axis as positive displacement, below as negative.
• A horizontal line means constant velocity (zero acceleration).
• The intercept at t = 0 gives initial velocity v_i.

Acceleration-vs-time (a-vs-t) graph:

• Area under the curve = change in velocity Δv.
• A horizontal line means constant acceleration.

Why this matters: Many exam problems present motion as a graph rather than giving you numbers. If you can read slopes and areas from these graphs, you can extract position, velocity, and acceleration without memorizing which equation to use. The graphs are the physics.

Proportional Reasoning

Kinematic relationships reveal powerful proportionalities:

• From Δx = v_i t + 12at² (with v_i = 0): displacement is proportional to the square of time. Double the time → quadruple the displacement (for constant acceleration from rest).
• From v_f² = v_i² + 2aΔx (with v_i = 0): final speed is proportional to the square root of displacement. Quadruple the distance → double the final speed.
• This square-root relationship explains why braking distance quadruples when speed doubles.

Limiting Cases

Test your understanding by asking: what happens when a variable goes to zero?

• What happens if a = 0? The kinematic equations reduce to Δx = v_i t — constant-velocity motion (Newton's first law, previewing Topic 2).
• What happens if v_i = 0? The equations simplify: Δx = 12at² and v_f² = 2aΔx. This describes an object starting from rest.
• What happens if g → 0? Projectiles would travel in straight lines forever. This is the deep-space limit."

Velocity-Time Graphs

On a v-vs-t graph:

• Slope = acceleration.
• Area under the curve = displacement Δx.
• A horizontal line means constant velocity (zero acceleration).
• A straight sloping line means constant acceleration.

ELI-10: Explain It Like I'm 10

You are in a car. If the speedometer needle stays at 30 mph, your acceleration is zero. If you press the gas and the needle climbs, you have positive acceleration. If you hit the brakes and the needle drops, you are accelerating in the opposite direction of your motion. Acceleration is just how quickly your velocity is changing. You can be moving very fast but have zero acceleration — just keep the pedal steady.

Common Mistakes

• "Acceleration always means speeding up." No — slowing down is also acceleration, just in the direction opposite to motion.
• "Zero velocity means zero acceleration." No — a ball thrown upward has zero velocity at the top of its path, but its acceleration is still g downward the entire time.

Key Takeaway

Acceleration is the slope of a velocity graph. The area under a velocity graph gives displacement. Acceleration can be positive, negative, or zero independently of velocity.

1.3 Constant Acceleration and the Kinematic Equations

Core Idea

When acceleration is constant, four equations link the five quantities v_i, v_f, a, Δx, Δt. You can solve any constant-acceleration problem by identifying which of the five you know and which you need.

Physics and Mathematics

The four kinematic equations (valid only when a is constant):

v_f = v_i + a t

Δx = v_i t + 12 a t²

v_f² = v_i² + 2a Δx

Δx = 12(v_i + v_f) t

Where:

• v_i = initial velocity (m/s)
• v_f = final velocity (m/s)
• a = constant acceleration (m/s²)
• Δx = displacement (m)
• t = time interval (s)

Choosing the right equation: Look at the knowns and the unknown. If time is not involved, use equation (3). If final velocity is not involved, use equation (2). Practice is the best teacher here.

Worked Example: Braking Distance

Problem: A car traveling at 20 m/s brakes with a constant deceleration of 5.0 m/s². How far does it travel before stopping?

Given: v_i = 20 m/s, v_f = 0, a = -5.0 m/s²

Find: Δx

Physics Principle: Constant acceleration with unknown time → use equation (3).

Equation: v_f² = v_i² + 2a Δx

Solution: 0 = (20)² + 2(-5.0)Δx 0 = 400 - 10Δx Δx = 40 m

Answer: 40 meters.

Does the Answer Make Sense? At highway speed (~45 mph), 40 m (about 10 car lengths) is a reasonable stopping distance. The answer is positive as expected for displacement in the direction of initial motion.

ELI-10: What Just Happened?

The car had lots of speed-energy when the driver hit the brakes. The brakes provided a steady slowing force. Equation (3) is the shortcut that skips time — it directly links the starting speed, the braking strength, and the stopping distance. Double the starting speed, and the stopping distance becomes four times as long. That is why highway speeds are so dangerous.

Free Fall

For objects in free fall near Earth's surface, a = -g = -9.8 m/s² (taking upward as positive). The kinematic equations work identically — just replace a with -g.

Key insight: In free fall, all objects (ignoring air resistance) accelerate downward at the same rate regardless of mass. A bowling ball and a marble dropped simultaneously will hit the ground together.

Worked Example: Ball Thrown Upward

Problem: A ball is thrown straight upward at 15 m/s from ground level. Find (a) the time to reach maximum height, (b) the maximum height, and (c) the total time in the air before returning to ground.

Given: v_i = +15 m/s, a = -9.8 m/s², v_f = 0 at top.

Find: t_up, y_max, t_total

Physics Principle: Constant acceleration (gravity). Use kinematic equations.

Solution (a): v_f = v_i + at → 0 = 15 + (-9.8)t → t = 1.53 s.

Solution (b): y_max = v_i t + 12at² = (15)(1.53) + 12(-9.8)(1.53)² ≈ 11.5 m. Alternatively, using v_f² = v_i² + 2aΔy: 0 = 225 + 2(-9.8)y_max → y_max = 11.5 m.

Solution (c): Symmetry — time up equals time down → t_total = 2 × 1.53 ≈ 3.06 s.

Does the Answer Make Sense? 1.5 seconds to rise 11.5 m — about the height of a 3-story building — from a strong throw is physically reasonable. The symmetry (time up = time down) holds only when launch and landing are at the same height with no air resistance.

ELI-10: What Just Happened?

Throw a ball straight up. It slows down as it rises because gravity pulls it downward. At the very top, it pauses for an instant (zero velocity), but gravity is still pulling — acceleration never becomes zero. Then it falls back down, speeding up at the same rate it slowed down. The climb and the fall each take the same amount of time, and the speed when it returns to your hand is the same as the speed you threw it (just downward now).

ELI-10: Explain It Like I'm 10

Gravity pulls everything downward with the same "oomph per kilogram." A heavy object feels a bigger gravitational pull, but it also has more mass to resist being sped up. These two effects cancel perfectly, so heavy and light things fall side by side — unless air gets in the way.

Common Mistakes

• Using kinematic equations when acceleration is NOT constant.
• Mixing sign conventions: pick a direction as positive at the start and stick with it.
• Forgetting that v_i = 0 at the top of a vertical throw does NOT mean a = 0 there.

Key Takeaway

Four equations, one condition: constant acceleration. Identify knowns, choose the equation that includes your unknown, and solve. Carry signs consistently.

1.4 Vectors

Core Idea

Many quantities in physics have both magnitude and direction. Vectors represent these quantities mathematically. Scalars have only magnitude.

Important Definitions

• Scalar: a quantity with magnitude only (mass, temperature, speed, distance, time).
• Vector: a quantity with magnitude and direction (displacement, velocity, acceleration, force).
• Components: the projections of a vector onto the x- and y-axes.

Physics and Mathematics

A vector A with magnitude A at angle θ from the +x axis has components:

A_x = A cosθ,   A_y = A sinθ

The magnitude and direction can be recovered:

A = √A_x² + A_y²,   θ= tan^-1(A_yA_x)

Vector addition: Add components separately: A + B has components (A_x + B_x, A_y + B_y).

ELI-10: Explain It Like I'm 10

A vector is an arrow. The arrow's length tells you "how much." The direction the arrow points tells you "which way." If you walk 3 blocks east and 4 blocks north, you end up 5 blocks northeast of where you started. Adding arrows means putting them tip-to-tail and drawing a new arrow from the start of the first to the tip of the last.

Common Mistakes

• Adding vector magnitudes directly (3 + 4 = 7) instead of using components or geometry (√(3² + 4²) = 5).
• Forgetting to specify direction when giving a vector answer.

Key Takeaway

Resolve vectors into perpendicular components. Perform algebra on components independently. Recombine at the end if needed.

1.5 Projectile Motion

Core Idea

Projectile motion is the combination of constant-velocity horizontal motion and constant-acceleration vertical motion (gravity). The horizontal and vertical motions are independent.

Important Definitions

• Launch velocity v₀ at angle θ: v_0x = v₀cosθ, v_0y = v₀sinθ.
• Time of flight: total time the projectile is in the air.
• Maximum height: the highest vertical position reached.
• Range R: horizontal distance traveled before landing at the same height.

Physics and Mathematics

Horizontal motion (constant velocity, a_x = 0): x = v_0x t

Vertical motion (constant acceleration, a_y = -g): y = v_0yt - 12gt²,   v_y = v_0y - gt

Time of flight (landing at same height): t_flight = 2v_0yg

Maximum height: y_max = v_0y²2g

Range: R = v₀² sin(2θ)g

Key result: Maximum range occurs at θ= 45^° for level-ground launches.

Worked Example: Soccer Kick

Problem: A soccer ball is kicked at 20 m/s at 30° above horizontal. Find its range.

Given: v₀ = 20 m/s, θ= 30^°, g = 9.8 m/s²

Find: Range R

Solution: R = v₀² sin(2θ)g = (20)² sin(60^°)9.8 = 400 × 0.8669.8 ≈ 35.4 m

Answer: About 35 meters.

Does the Answer Make Sense? A kicked soccer ball traveling ~35 m (over a third of a football field) at 20 m/s is physically reasonable. The units check: m²/s² divided by m/s² gives meters.

ELI-10: Explain It Like I'm 10

Imagine throwing a ball. The ball moves forward because you threw it forward — nothing is pushing it forward once it leaves your hand (ignoring air). Meanwhile, gravity pulls it down the whole time. The forward motion and the downward motion happen independently. The ball stays in the air longer if you throw it more upward, but it goes farthest forward if you split the throw evenly between forward and upward — that is the 45-degree sweet spot.

Common Mistakes

• Thinking the horizontal velocity changes during flight (it does not, absent air resistance).
• Treating the motion as a single 2D problem instead of two independent 1D problems.
• Forgetting that v_y = 0 at the top of the trajectory (only the vertical component is zero; horizontal velocity continues).

Key Takeaway

Projectile motion = independent horizontal (constant v) + vertical (constant a = -g) motions. Solve each direction separately using the same time t.

1.6 Uniform Circular Motion

Core Idea

An object moving in a circle at constant speed is accelerating because its direction is continuously changing. The acceleration points toward the center.

Physics and Mathematics

Centripetal acceleration: a_c = v²r

Where:

• a_c = centripetal acceleration (m/s²), directed toward the center
• v = tangential speed (m/s)
• r = radius of the circle (m)

Period T = time for one full revolution: v = 2πrT

ELI-10: Explain It Like I'm 10

Tie a ball to a string and swing it in a circle over your head. The ball is moving at a steady speed, but it is always changing direction. Acceleration is any change in velocity, and velocity includes direction — so the ball is accelerating even though its speed never changes. The acceleration points inward, toward your hand, because that is the direction the velocity is turning.

Common Mistakes

• "Uniform circular motion has zero acceleration because the speed is constant." No — velocity is a vector; changing direction means changing velocity, which means acceleration.
• Confusing centripetal (center-seeking) with centrifugal (outward-feeling). The actual force/acceleration on the object is inward.

Key Takeaway

Uniform circular motion involves acceleration toward the center with magnitude v²/r. Constant speed does not mean zero acceleration when the path curves.

1.7 Relative Velocity

Core Idea

Velocity is measured relative to a frame of reference. Two observers in different frames may measure different velocities for the same object.

Physics and Mathematics

If object P has velocity v_PA relative to frame A, and frame A has velocity v_AB relative to frame B, then:

v_PB = v_PA + v_AB

Example: A passenger walks forward at 1 m/s inside a train moving at 30 m/s relative to the ground. The passenger's velocity relative to the ground is 31 m/s forward.

ELI-10: Explain It Like I'm 10

You are sitting on a moving train. To someone on the train, you are not moving. To someone standing outside watching the train go by, you are moving at the train's speed. If you then walk down the aisle, the person outside sees you moving at train-speed plus walking-speed. "How fast" always depends on "compared to what."

Common Mistakes

• Adding speeds without considering direction (vector addition, not just arithmetic).
• Assuming there is one true or absolute velocity.

Key Takeaway

All velocities are relative to a chosen reference frame. Galilean velocity addition works for everyday speeds: v_PB = v_PA + v_AB.

Topic Summary

• Position locates an object; displacement is the vector change in position; distance is the scalar path length.
• Velocity is the rate of change of position; speed is its magnitude.
• Acceleration is the rate of change of velocity; it can mean speeding up, slowing down, or changing direction.
• For constant acceleration, four kinematic equations relate v_i, v_f, a, Δx, t.
• Free fall means acceleration is g ≈ 9.8 m/s² downward, independent of mass.
• Vectors have magnitude and direction; add them by resolving into components.
• Projectile motion separates into independent horizontal (constant v) and vertical (constant a = -g) motions.
• Uniform circular motion has acceleration v²/r toward the center even at constant speed.
• Relative velocity depends on the observer's frame of reference.

Essential Equations

Concept Check

1. Can an object have zero velocity and nonzero acceleration simultaneously? Give an example.
2. A ball is thrown straight up. At its highest point, what is its velocity? What is its acceleration?
3. Does a car's speedometer measure speed or velocity? Why does this distinction matter when driving on a curved road?
4. If you double the launch speed of a projectile, by what factor does its maximum height change? (Assume same launch angle.)
5. An object moves in a circle at constant speed. Is any work being done on it by the centripetal force? (Hint: Topic 3.)

Open Educational References

• OpenStax, College Physics, Chapter 2: Kinematics
• OpenStax, University Physics, Volume 1, Chapters 3–4: Motion in 1D, 2D, and 3D