Velocity and Acceleration
Understand the difference between speed and velocity, explore instantaneous and average values, and interpret velocity-time graphs to analyse motion.
Speed vs Velocity
Speed is a scalar quantity -- it tells you how fast an object is moving, but not the direction. Velocity is a vector quantity -- it includes both magnitude (speed) and direction. In physics, this distinction is critical.
Average Velocity
vavg = Δx / Δt
The total displacement divided by the total time taken. Displacement is the straight-line distance from start to finish, with direction.
Instantaneous Velocity
v = limΔt→0 Δx/Δt
The velocity at a specific instant in time. On a position-time graph, it equals the gradient of the tangent at that point.
Key distinction: A car travelling around a circular track at a constant speed of 60 km/h has a constantly changing velocity because its direction is always changing.
Acceleration
Acceleration is the rate of change of velocity with respect to time. It is a vector quantity measured in metres per second squared (m s-2). An object accelerates whenever its speed or direction changes.
Equations of Uniform Acceleration (SUVAT)
v = u + at
Final velocity from initial velocity and acceleration
s = ut + ½at²
Displacement from initial velocity, acceleration and time
v² = u² + 2as
Final velocity without needing time
s = ½(u + v)t
Displacement from average of initial and final velocities
Where: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time
Remember: Negative acceleration (deceleration) means the object is slowing down in its direction of motion. Acceleration due to gravity near Earth's surface is approximately 9.8 m s-2 downward.
Velocity-Time Graphs
Velocity-time (v-t) graphs provide a powerful visual tool for analysing motion. The gradient of the graph gives the acceleration, and the area under the curve gives the displacement.
Reading a Velocity-Time Graph
Positive gradient (line slopes up)
Object is accelerating -- velocity increasing
Zero gradient (horizontal line)
Constant velocity -- no acceleration
Negative gradient (line slopes down)
Object is decelerating -- velocity decreasing
Area under the graph
Equals the displacement of the object
Interpreting Graph Shapes
Straight line from origin: Uniform acceleration from rest -- the velocity increases at a constant rate.
Curved line: Non-uniform (changing) acceleration -- the rate of velocity change itself is changing.
Line below the time axis: The object is moving in the negative direction (opposite to the chosen positive direction).
Key Vocabulary
Displacement
The change in position of an object, measured as a straight-line distance with direction from start to finish. SI unit: metres (m).
Velocity
The rate of change of displacement with respect to time. A vector quantity with SI unit m s-1.
Acceleration
The rate of change of velocity with respect to time. SI unit: m s-2. Can be positive (speeding up) or negative (slowing down).
Uniform Motion
Motion at constant velocity (zero acceleration). On a v-t graph, this appears as a horizontal line.
Worked Examples
A car accelerates uniformly from 10 m s-1 to 30 m s-1 in 5 seconds. Calculate the acceleration.
Step 1: Identify known values: u = 10 m s-1, v = 30 m s-1, t = 5 s.
Step 2: Use v = u + at, rearranged to a = (v - u) / t.
Step 3: a = (30 - 10) / 5 = 20 / 5 = 4 m s-2.
Answer: The car accelerates at 4 m s-2.
A cyclist travels 400 m north in 50 s, then 300 m south in 30 s. Calculate the average speed and average velocity.
Step 1: Total distance = 400 + 300 = 700 m. Total time = 50 + 30 = 80 s.
Step 2: Average speed = total distance / total time = 700 / 80 = 8.75 m s-1.
Step 3: Total displacement = 400 north - 300 south = 100 m north.
Step 4: Average velocity = displacement / time = 100 / 80 = 1.25 m s-1 north.
A car brakes uniformly from 20 m s-1 to rest over a distance of 40 m. Find the deceleration.
Step 1: Known: u = 20 m s-1, v = 0, s = 40 m.
Step 2: Use v² = u² + 2as ⇒ 0 = (20)² + 2a(40).
Step 3: 0 = 400 + 80a ⇒ a = -400/80 = -5 m s-2.
Answer: The deceleration is 5 m s-2 (acceleration = -5 m s-2).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the SI unit of acceleration?
Question 2
A train accelerates uniformly from rest to 25 m s-1 in 10 s. What is its acceleration?
Question 3
On a velocity-time graph, what does the area under the line represent?
Question 4
Which quantity is a vector?
Question 5
A ball is dropped from rest and falls for 3 s (g = 9.8 m s-2). What is its final velocity?
Key Concepts Summary
- ●Speed is scalar (magnitude only); velocity is vector (magnitude + direction).
- ●Average velocity = total displacement / total time; instantaneous velocity is the velocity at a single moment.
- ●Acceleration = (v - u) / t, measured in m s-2.
- ●The four SUVAT equations apply to uniform acceleration problems.
- ●On a v-t graph, the gradient = acceleration and the area under the curve = displacement.