Difficulty: Introductory | Prerequisites: Basic algebra and trigonometry (sine, cosine, tangent). Familiarity with coordinate systems and graphs is helpful.
This is the bread and butter of introductory mechanics. Kinematics describes how objects move, vectors let you handle motion in two dimensions, and projectile motion combines both with gravity. These topics appear on virtually every PHYS 114 exam and form the foundation for everything that follows in the course, from Newton's laws to energy conservation. If you skipped the first few weeks, start here.
Kinematics is the study of motion without worrying about what causes it. You learn to read and interpret position-time and velocity-time graphs, break movement into components using vectors, and predict the path of objects launched at an angle (projectile motion). Free-fall problems are a special case where the only acceleration is gravity. Mastering these tools is essential before moving on to Newton's laws, because you will need them to describe what happens once forces enter the picture.
Kinematics uses graphs and equations to describe motion in terms of position, velocity, and acceleration. Vectors let you add displacements and velocities that point in different directions. Projectile motion splits a launched object into independent horizontal (constant velocity) and vertical (constant acceleration due to gravity) components.
Position (x)
The location of an object along a chosen axis, measured from an origin. Think of it as "where the object is" at a given moment.
Displacement
The change in position, a vector quantity with both magnitude and direction: Δx = x_final − x_initial. In simple terms, displacement tells you how far and in what direction you ended up from where you started, regardless of the route you took.
Velocity (v)
The rate of change of position with respect to time. It is a vector, so it has direction. In simple terms, velocity is speed with a direction attached.
Instantaneous velocity
The velocity at a single moment in time, equal to the slope of the position-time graph at that point. Think of it as what your speedometer reads at one precise instant.
Acceleration (a)
The rate of change of velocity with respect to time. In simple terms, acceleration tells you how quickly your velocity is changing, and in which direction.
Free fall
Motion under the influence of gravity alone, with no air resistance. The acceleration is g = 9.8 m/s² downward, regardless of the object's mass.
Projectile motion
The two-dimensional motion of an object launched at an angle, subject only to gravitational acceleration. Think of it as free fall with a sideways component that keeps going at constant speed.
Vector
A quantity that has both magnitude and direction. Displacement, velocity, and acceleration are all vectors. In simple terms, a vector tells you "how much" and "which way."
Scalar
A quantity that has magnitude only, with no direction. Speed, distance, and time are scalars.
Component (of a vector)
The projection of a vector along a coordinate axis. For a vector of magnitude A at angle θ from the +x axis: Ax = A cos θ and Ay = A sin θ. Think of components as the shadow a vector casts along each axis.
A position-time (x vs t) graph shows where an object is at each moment.
The slope of the graph at any point equals the instantaneous velocity.
A straight line with positive slope = constant positive velocity (moving in the +x direction).
A straight line with negative slope = constant negative velocity (moving in the −x direction).
A horizontal line (zero slope) = the object is stationary.
A curved line = the velocity is changing, meaning the object is accelerating.
When the graph has segments with different slopes, the velocity changes abruptly at the transition points.
Example: if position increases linearly from t = 0 to t = 1 s, stays constant from t = 1 to t = 3 s, then decreases linearly from t = 3 to t = 4 s, the velocity at t = 2 s is 0 m/s (the flat segment).
Reading the slope correctly is the single most important skill for this type of question. Rise over run, with attention to sign.
To find net displacement when an object moves along two or more legs at different angles, break each leg into x and y components, then add components separately.
For a vector of magnitude A at angle θ (measured counter-clockwise from the +x axis): x-component = A cos θ, y-component = A sin θ.
Once you have total x and total y components, find the magnitude of the resultant: |R| = √(Rx² + Ry²).
The direction of the resultant is found from: θ = arctan(Ry / Rx), with attention to the correct quadrant.
Example: a player runs 10.0 m at 50.0° ccw from +x, then jogs 13.0 m at 155.0° ccw from +x.
First leg: x₁ = 10.0 cos 50° ≈ 6.43 m, y₁ = 10.0 sin 50° ≈ 7.66 m.
Second leg: x₂ = 13.0 cos 155° ≈ −11.78 m, y₂ = 13.0 sin 155° ≈ 5.49 m.
Resultant: Rx = 6.43 + (−11.78) = −5.35 m, Ry = 7.66 + 5.49 = 13.15 m.
Magnitude: √(5.35² + 13.15²) ≈ 14.2 m.
A projectile is any object moving through the air under gravity alone (no air resistance).
The key insight: horizontal and vertical motions are independent of each other.
Horizontal: no acceleration, so vx = v₀ cos θ throughout the flight. The horizontal component never changes.
Vertical: constant acceleration of g = 9.8 m/s² downward. The vertical component changes exactly as in free fall.
At the peak of a projectile's path, the vertical velocity is zero. The object is still moving horizontally.
This is why, when the archerfish problem says the water is "moving perfectly horizontally" at the target, the vertical component has reached zero, meaning the target is at the peak. The horizontal component is v₀ cos θ.
Time of flight on level ground: the ball returns to its launch height when the vertical displacement is zero. Using y = v₀y t − ½g t² = 0, solve for t = 2 v₀ sin θ / g.
Example: v₀ = 46.5 m/s at 37.5° gives t = 2(46.5)(sin 37.5°) / 9.8 ≈ 5.77 s.
The horizontal component of initial velocity: vx = v₀ cos θ.
Example: v₀ = 2.30 m/s at 19.5° gives vx = 2.30 cos 19.5° ≈ 2.17 m/s.
On a velocity-time (v vs t) graph, the area under the curve equals the displacement.
For a straight line from (0, 0) to (5, 40), the area is a triangle: ½ × 5 × 40 = 100 m.
For a straight line from (0, 10) to (5, 30), the area is a trapezoid: ½ × (10 + 30) × 5 = 100 m.
For a horizontal line at v = 20 m/s from t = 0 to t = 5, the area is a rectangle: 20 × 5 = 100 m.
All three of these shapes give the same displacement (100 m), which is why, in the drag racer problem, all three cars travel the same distance.
The slope of a velocity-time graph equals the acceleration.
Positive slope = speeding up (if velocity is positive).
Zero slope = constant velocity.
In free fall, the only acceleration is g = 9.8 m/s² directed downward.
An object dropped from rest falls a distance given by: d = ½ g t², so t = √(2d / g).
Example: a water drop falling 4.0 m from a stalactite takes t = √(2 × 4.0 / 9.8) ≈ 0.904 s.
All objects in free fall accelerate at the same rate, regardless of mass (assuming no air resistance).
The sign convention matters: if you define "up" as positive, then a = −9.8 m/s². If "down" is positive, a = +9.8 m/s². Be consistent within a problem.
Strobe diagrams (motion diagrams) show the position of an object at equal time intervals.
Increasing spacing between dots = the object is speeding up.
Decreasing spacing = the object is slowing down.
Equal spacing = constant velocity.
Constant-velocity motion: x = x₀ + v t
Constant-acceleration kinematics (the "big four"):
v = v₀ + a t
x = x₀ + v₀ t + ½ a t²
v² = v₀² + 2 a (x − x₀)
x = x₀ + ½ (v₀ + v) t
Free-fall substitution: replace a with g (9.8 m/s² downward)
Projectile motion components:
Horizontal: x = v₀ cos θ · t (no acceleration)
Vertical: y = v₀ sin θ · t − ½ g t²
Vector magnitude: |R| = √(Rx² + Ry²)
Vector direction: θ = arctan(Ry / Rx)
Time of flight (level ground): T = 2 v₀ sin θ / g
Range (level ground): R = v₀² sin 2θ / g
Projectile motion is how sports analysts model the trajectory of a golf ball, a basketball, or a punted football. The same decomposition into horizontal and vertical components is used by engineers designing irrigation sprinklers and by the archerfish, which squirts water at insects above the surface. Velocity-time graph analysis is the basis of accelerometer data processing in phones and fitness trackers.
Students often think that at the peak of a projectile's path the velocity is zero. The vertical component is zero, but the horizontal component remains v₀ cos θ. The object is still moving.
Students confuse distance and displacement. Distance is the total path length (a scalar), while displacement is the straight-line change in position (a vector). They are only equal for motion in one direction without backtracking.
Students sometimes think acceleration means "going fast." An object can be momentarily at rest and still have a large acceleration (for example, a ball at the peak of its arc is still accelerating downward at g).
Students forget sign conventions when working with free fall. If you define up as positive, g enters the equations as −9.8 m/s², not +9.8 m/s².
⚠️ Reading the slope of a position-time graph is a near-certain exam question. Know what each segment's slope means.
⚠️ Vector addition by components comes up in multiple-choice and short-answer forms. Practise breaking vectors into x and y parts until it is automatic.
⚠️ The independence of horizontal and vertical motion in projectile problems is tested constantly. Expect to be asked for the horizontal component of velocity, time of flight, or peak height.
⚠️ Free-fall timing problems (like the stalactite drip) require careful setup of d = ½ g t². Watch your units.
⚠️ Velocity-time graph area problems (like the three drag racers) test whether you understand that displacement = area under the v-t curve.
True or false: the slope of a position-time graph gives the acceleration. (False, it gives velocity.)
Fill in the blank: in projectile motion, the horizontal velocity ______ throughout the flight. (remains constant)
True or false: two vectors of the same magnitude always produce the same resultant. (False, direction matters.)
Fill in the blank: the area under a velocity-time graph equals the ______. (displacement)
True or false: an object in free fall at its highest point has zero acceleration. (False, acceleration is still g downward.)
Q: An object's position-time graph is flat (horizontal) between t = 1 s and t = 3 s. What is the instantaneous velocity at t = 2 s?
A: 0 m/s. A flat segment means the position is not changing, so the slope (velocity) is zero.
Q: A hiker walks 10.0 m at 50° north of east, then 13.0 m at 155° counter-clockwise from east. What is the magnitude of the net displacement?
A: About 14.2 m. Break each leg into components, sum the x-components and y-components separately, then use the Pythagorean theorem.
Q: An archerfish squirts water at 2.30 m/s at 19.5° above horizontal. What is the horizontal component of the initial velocity?
A: vx = 2.30 cos 19.5° ≈ 2.17 m/s. The horizontal component does not change during flight.
Q: A golfer hits a ball at 46.5 m/s at 37.5° above horizontal on level ground. How long is the ball in the air?
A: T = 2(46.5)(sin 37.5°) / 9.8 ≈ 5.77 s. Use the time-of-flight formula for level ground.
Q: A stalactite drips water from 4.0 m above a pool. How long does a single drop take to fall? (g = 9.8 m/s²)
A: t = √(2 × 4.0 / 9.8) ≈ 0.90 s. Use d = ½ g t² with the drop starting from rest.
Q: Three drag racers have v-t graphs: Andy from (0,0) to (5,40), Betty from (0,10) to (5,30), Carl at v = 20 m/s. At t = 5 s, who has travelled farthest?
A: All have travelled the same distance (100 m). Each area under the curve equals 100 m.
Kinematics feeds directly into Newton's second law (F = ma), because once you can describe acceleration, you can relate it to force. Projectile motion reappears in energy conservation problems: the kinetic and potential energy at different points along a trajectory must obey conservation laws. Vector decomposition is also the basis for resolving forces into components on free-body diagrams, which you will use throughout the rest of the course.
kinematics, position-time graph, x-t graph, velocity-time graph, v-t graph, displacement, distance, speed vs velocity, acceleration, free fall, projectile motion, range equation, time of flight, vector addition, vector components, resultant vector, strobe diagram, motion diagram, PHYS 114, general physics, introductory mechanics, University of Washington