Kinematics
Grade 11 · Science · Study Guide
Study the motion of objects — describing displacement, velocity, and acceleration using graphs and equations without worrying about the forces that cause them.
Kinematics is the branch of mechanics that describes motion without analyzing the forces that cause it. Four key quantities define any straight-line motion: displacement (Δx), velocity (v), acceleration (a), and time (t). Displacement is the change in position — a vector quantity with both magnitude and direction. Distance is the total path length — a scalar. Average velocity equals displacement divided by time; average speed equals total distance divided by time.
Acceleration is the rate of change of velocity. An object accelerates when it speeds up, slows down, or changes direction. For uniform (constant) acceleration, four kinematic equations relate the five variables: v = v₀ + at; Δx = v₀t + ½at²; v² = v₀² + 2aΔx; Δx = (v + v₀)t/2. These equations are derived from the definitions of velocity and acceleration and assume constant acceleration throughout the interval.
Graphical analysis is a powerful tool. On a position-time (x-t) graph, slope equals velocity; a curved line indicates acceleration. On a velocity-time (v-t) graph, slope equals acceleration and the area under the curve equals displacement. These relationships hold for all uniformly accelerating systems.
Projectile motion occurs when an object is launched at an angle and moves under gravity alone (no air resistance). The key insight is that horizontal and vertical motions are completely independent. Horizontal velocity is constant (no horizontal acceleration), while vertical motion is uniformly accelerated by gravity (g = 9.8 m/s² downward). Time of flight is determined by the vertical equations alone. Range (horizontal distance) = v₀cosθ × t. Maximum range for a given launch speed occurs at 45°.
| Equation | Variables Used | Missing Variable |
|---|---|---|
| v = v₀ + at | v, v₀, a, t | Δx |
| Δx = v₀t + ½at² | Δx, v₀, a, t | v |
| v² = v₀² + 2aΔx | v, v₀, a, Δx | t |
| Δx = (v + v₀)t/2 | Δx, v, v₀, t | a |
| Graph Type | Slope Represents | Area Represents |
|---|---|---|
| Position-Time (x vs t) | Velocity | — |
| Velocity-Time (v vs t) | Acceleration | Displacement |
| Acceleration-Time (a vs t) | Rate of acceleration change | Change in velocity |
| Component | Acceleration | Initial Velocity | Key Equations |
|---|---|---|---|
| Horizontal (x) | 0 (constant velocity) | v₀cosθ | x = v₀cosθ × t |
| Vertical (y) | −g = −9.8 m/s² | v₀sinθ | y = v₀sinθ·t − ½gt² |
Key terms
- Displacement (Δx)
- Change in position from initial to final point; vector with magnitude and direction.
- Velocity (v)
- Rate of change of displacement; v = Δx/Δt. Vector quantity.
- Acceleration (a)
- Rate of change of velocity; a = Δv/Δt. Vector quantity. Negative acceleration = deceleration if in same direction as motion.
- Uniform Acceleration
- Constant acceleration throughout a time interval; the four kinematic equations apply.
- Projectile Motion
- 2D motion under gravity alone; horizontal and vertical components are independent.
- Free Fall
- Motion under gravity only; a = g = 9.8 m/s² downward (ignoring air resistance).
Practice questions
A car starts from rest and accelerates at 3.0 m/s² for 5.0 s. What is its final velocity?
- 8.0 m/s
- 15 m/s
- 37.5 m/s
- 45 m/s
Show answer
v = v₀ + at = 0 + (3.0)(5.0) = 15 m/s.
How far does the same car travel during those 5.0 s?
- 15 m
- 37.5 m
- 75 m
- 150 m
Show answer
Δx = v₀t + ½at² = 0 + ½(3.0)(25) = 37.5 m.
On a velocity-time graph, the slope of the line equals the object's ______.
Show answer
Slope = rise/run = Δv/Δt = acceleration.
A ball is dropped from rest. Ignoring air resistance, how fast is it moving after 3.0 s? (g = 9.8 m/s²)
- 9.8 m/s
- 19.6 m/s
- 29.4 m/s
- 44.1 m/s
Show answer
v = v₀ + at = 0 + (9.8)(3.0) = 29.4 m/s.
In projectile motion, which component of velocity remains constant (no air resistance)?
- Vertical component
- Horizontal component
- Both components
- Neither component
Show answer
There is no horizontal force in projectile motion (ignoring air resistance), so horizontal velocity remains constant throughout the flight.
Fun facts
- Galileo disproved the ancient idea that heavier objects fall faster by (reportedly) dropping two different-mass spheres from the Leaning Tower of Pisa — both hit the ground simultaneously.
- A bullet fired horizontally and one dropped from the same height hit the ground at exactly the same time — a perfect demonstration that horizontal and vertical motions are independent.
- The world record for the 100 m sprint (Usain Bolt, 9.58 s) involves a peak acceleration of about 4.5 m/s² at the start — close to half of gravitational acceleration.
Get a personalized course on Kinematics built around your child.