Unit 1 — Kinematics
Describing motion without asking what causes it: scalars and vectors, displacement, velocity, acceleration, graphs, relative motion, and projectiles. Each section ends with a short practice set — click "Show answer" to check yourself.
Scalars and Vectors in One Dimension
A scalar has magnitude (size) only. A vector has both magnitude and direction. In one dimension, direction is shown with a sign: choose a positive direction (e.g. right or up) and treat the opposite as negative.
| Scalar | Vector |
|---|---|
Distance (d) | Displacement (Δx) |
| Speed | Velocity (v) |
| Time, mass, energy | Acceleration (a), force |
- Distance is the total length of the path travelled; it is never negative and never decreases.
- Displacement is the straight-line change in position, from start to finish, with direction.
- The magnitude of a vector is always positive; the sign only encodes direction.
1. A student walks 5 m east, then 3 m west. Find the distance travelled and the displacement (take east as positive).
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2. Classify each as scalar or vector: 30 m/s, 30 m/s north, 12 kg, −4 m/s².
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Displacement, Velocity, and Acceleration
These three quantities describe how position changes with time.
| Quantity | Definition | Units |
|---|---|---|
| Displacement | Δx = xf − xi | m |
| Average velocity | vavg = Δx / Δt | m/s |
| Average speed | distance / Δt | m/s |
| Average acceleration | aavg = Δv / Δt | m/s² |
For motion with constant acceleration in one dimension, use the kinematic equations:
v = v₀ + a·t Δx = v₀·t + ½·a·t² v² = v₀² + 2·a·Δx
- If v and a have the same sign, the object speeds up; if opposite signs, it slows down.
- An object can have zero velocity and non-zero acceleration (a ball at the top of its flight).
- Free fall near Earth's surface (ignoring air resistance):
a = −g ≈ −9.8 m/s²(down), whether the object is rising or falling.
1. A car speeds up from 10 m/s to 30 m/s in 5.0 s. What is its average acceleration?
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a = Δv/Δt = (30 − 10)/5.0 = 4.0 m/s².2. A ball is thrown straight up. At its highest point, what are its velocity and acceleration?
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Representing Motion
Motion can be shown with words, equations, motion diagrams (dots at equal time intervals), and graphs. Graphs are the most tested form.
| Graph | Slope gives | Area under curve gives |
|---|---|---|
| Position vs. time (x–t) | Velocity | — |
| Velocity vs. time (v–t) | Acceleration | Displacement |
| Acceleration vs. time (a–t) | — | Change in velocity |
- x–t: a straight line means constant velocity; a curve means changing velocity; a flat line means at rest.
- v–t: a horizontal line means constant velocity (a = 0); a straight sloped line means constant acceleration.
- On a v–t graph, area above the axis is positive displacement and area below is negative.
- Motion diagrams: evenly spaced dots mean constant velocity; dots spreading apart mean speeding up; dots bunching together mean slowing down.
1. An object's v–t graph rises in a straight line from 0 to 6 m/s over 3 s, then stays at 6 m/s for 4 s. Find its total displacement.
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2. On an x–t graph, the curve gets steeper as time passes. What does this say about the object's motion?
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Reference Frames and Relative Motion
Velocity is always measured relative to a reference frame, the observer or object you treat as at rest. A passenger sitting on a train is at rest relative to the train but moving relative to the ground.
v(A rel. C) = v(A rel. B) + v(B rel. C)
- In one dimension, add velocities with signs; the "B" terms cancel like fractions.
- Displacement and velocity depend on the frame; acceleration is the same in all frames moving at constant velocity relative to each other.
- Always state the frame explicitly: "relative to the ground," "relative to the water."
1. A train moves at 20 m/s east relative to the ground. A passenger walks at 2 m/s toward the rear (west) relative to the train. What is the passenger's velocity relative to the ground?
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2. A boat moves at 5 m/s relative to the water. The river flows at 3 m/s downstream. Find the boat's speed relative to the shore going (a) downstream and (b) upstream.
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Vectors and Motion in Two Dimensions
In two dimensions, break each vector into perpendicular components and treat x and y separately.
Aₓ = A·cos θ A_y = A·sin θ A = √(Aₓ² + A_y²) θ = tan⁻¹(A_y / Aₓ)
Projectile motion (only gravity acts) is 2D motion where the two directions are independent:
| Horizontal (x) | Vertical (y) | |
|---|---|---|
| Acceleration | 0 | −g (≈ −9.8 m/s²) |
| Velocity | Constant | Changes uniformly |
| Equation | x = vₓ·t | y = vy0·t − ½gt² |
- Time is the only link between the x and y equations.
- At the peak of the trajectory,
vy = 0butvₓis unchanged. - A ball dropped and a ball launched horizontally from the same height hit the ground at the same time.
- For launch and landing at equal heights, the range is greatest at 45° (and equal for complementary angles).
1. A ball is kicked at 20 m/s at 30° above the horizontal. Find its initial horizontal and vertical velocity components.
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2. A rock is thrown horizontally at 8.0 m/s from a 45 m high cliff. How long does it take to land, and how far from the base does it land? (Use g = 9.8 m/s².)