Cambridge AS & A Level Physics (9702) – Kinematics
Objective
To obtain instantaneous velocity and acceleration from displacement‑time (s‑t) and velocity‑time (v‑t) graphs, to relate average quantities to areas under graphs, and to use these graphical techniques to derive, understand and apply the five SUVAT equations for uniformly accelerated motion.
Learning Outcomes (Syllabus 2 Kinematics)
- Define distance, displacement, speed, velocity and acceleration.
- Interpret straight‑line and curved sections of s‑t, v‑t and a‑t graphs.
- Obtain instantaneous velocity from the gradient of an s‑t graph and instantaneous acceleration from the gradient of a v‑t graph.
- Calculate average velocity and average acceleration and relate them to the areas under the appropriate graphs.
- Derive and use the five SUVAT equations for uniformly accelerated motion.
- Apply the graphical method to real‑world situations (e.g. free fall, motion with air resistance).
- Estimate uncertainties when reading gradients or areas from plotted data (AO1 & AO2).
- Link constant acceleration to Newton’s second law (F = ma).
Key Definitions
| Quantity | Symbol | Definition |
|---|
| Distance | Δs (scalar) | Length of the path travelled; always positive. |
| Displacement | s (vector) | Change in position relative to a chosen origin; may be positive or negative. |
| Speed | \(\bar{v}\) (scalar) | Rate of change of distance; \(\bar{v}=Δs/Δt\). |
| Velocity | v (vector) | Rate of change of displacement; \(\bar{v}=Δs/Δt\) (average) or \(v=ds/dt\) (instantaneous). |
| Acceleration | a (vector) | Rate of change of velocity; \(\bar{a}=Δv/Δt\) (average) or \(a=dv/dt\) (instantaneous). |
Reading and Interpreting Graphs
1. Straight‑line sections
- s‑t graph: A straight line indicates constant velocity. Gradient = velocity.
- v‑t graph: A straight line indicates constant acceleration. Gradient = acceleration; area under the line = change in displacement.
- a‑t graph: A straight line indicates constant jerk (rate of change of acceleration). Gradient = jerk; area = change in velocity.
2. Curved sections
- On an s‑t graph, curvature means the velocity is changing – i.e. the object is accelerating.
- On a v‑t graph, a curve shows that acceleration is not constant. The steeper the curve, the larger the instantaneous acceleration at that point.
- On an a‑t graph, curvature indicates a varying acceleration (non‑zero jerk).
3. Instantaneous values – using a tangent
- Draw a tangent line at the point of interest on the graph.
- Instantaneous velocity \(v\) = gradient of the tangent on an s‑t graph.
- Instantaneous acceleration \(a\) = gradient of the tangent on a v‑t graph.
- Remember: the gradient of a *secant* line (joining two points) gives the average value, not the instantaneous one.
4. Average values – area method
- Average velocity over a time interval \(\Delta t\) is the total displacement divided by \(\Delta t\). Graphically it is the *gradient* of the straight line joining the two end‑points on an s‑t graph.
- Average acceleration over \(\Delta t\) is the change in velocity divided by \(\Delta t\). Graphically it is the *gradient* of the straight line joining the two end‑points on a v‑t graph.
- Displacement from a v‑t graph: Area under the curve between \(t_1\) and \(t_2\) equals \(\int_{t_1}^{t_2} v\,dt\) = total displacement.
- Change in velocity from an a‑t graph: Area under the curve between \(t_1\) and \(t_2\) equals \(\int_{t_1}^{t_2} a\,dt\) = \(\Delta v\).
Worked Example – Area under a v‑t graph
- Given a v‑t graph where velocity increases linearly from 0 m s\(^{-1}\) at \(t=0\) s to 10 m s\(^{-1}\) at \(t=5\) s (a straight line).
- The area is a right‑angled triangle: \(\text{Area}= \tfrac{1}{2}\times\text{base}\times\text{height}= \tfrac{1}{2}\times5\;\text{s}\times10\;\text{m s}^{-1}=25\;\text{m}\).
- Thus the displacement during the 5 s interval is 25 m.
Derivation of the SUVAT Equations (Uniform Acceleration)
Starting from the definitions of acceleration and velocity:
\[
a = \frac{dv}{dt}\qquad\text{and}\qquad v = \frac{ds}{dt}
\]
- First equation (definition of acceleration):
Integrate \(a = \frac{dv}{dt}\) with constant \(a\):
\[
\int_{v_0}^{v} dv = a\int_{t_0}^{t} dt \;\Rightarrow\; v = v_0 + at
\]
- Second equation (definition of velocity):
Integrate \(v = \frac{ds}{dt}\) using the result for \(v\):
\[
s = \int_{0}^{t} (v_0+at)\,dt = v_0t + \tfrac{1}{2}at^{2}
\]
- Third equation (eliminate time):
Solve the first equation for \(t = (v-v_0)/a\) and substitute into the second:
\[
s = v_0\left(\frac{v-v_0}{a}\right)+\tfrac{1}{2}a\left(\frac{v-v_0}{a}\right)^{2}
\]
Simplifying gives
\[
v^{2}=v_0^{2}+2as
\]
- Fourth equation (average velocity):
For constant acceleration the average velocity \(\bar{v}\) over time \(t\) is \(\frac{v_0+v}{2}\). Using \(s=\bar{v}t\):
\[
s = \frac{v_0+v}{2}\,t
\]
- Fifth equation (alternative form of the second):
Combine the first and fourth equations to eliminate \(v\):
\[
s = v_0t + \tfrac{1}{2}at^{2}
\]
(already obtained as equation 2, but retained for completeness in the exam syllabus.)
Graphical Application to Real‑World Situations
Free fall (no air resistance)
- Acceleration is constant: \(a = g = 9.81\;\text{m s}^{-2}\) downwards.
- v‑t graph: straight line through the origin with gradient \(-g\).
- Instantaneous velocity at any time \(t\): \(v = -gt\).
- Displacement after time \(t\): area under the line, \(s = \tfrac{1}{2}gt^{2}\) (downwards).
Free fall with linear air resistance
When the resistive force is proportional to velocity, \(F_{R}= -k v\), the net force is \(F = mg - kv\). The acceleration varies with time, giving a curved v‑t graph that asymptotically approaches a terminal velocity \(v_t = mg/k\).
- Initial part of the curve is steep (large acceleration); as speed increases, the gradient (acceleration) decreases.
- Graphical interpretation:
- Gradient of the tangent at any point = instantaneous acceleration.
- Area under the curve up to a given time = displacement.
Estimating Uncertainties from Graphs
- Gradient uncertainty: Use the “half‑grid” method. If the smallest vertical and horizontal divisions are \(\Delta v\) and \(\Delta t\), then the uncertainty in the gradient \(\delta a\) is approximately
\[
\delta a \approx \frac{\sqrt{(\Delta v)^{2}+(\Delta t)^{2}}}{\Delta t^{2}}
\]
(or simply \(\pm\frac{1}{2}\) grid‑square in each direction and propagate).
- Area uncertainty: Count the number of full grid squares under the curve; the uncertainty is \(\pm\frac{1}{2}\) square for each side of the counted region. Convert to physical units using the scale.
- Propagation to derived quantities: If \(v = s/t\), then
\[
\frac{\delta v}{v} = \sqrt{\left(\frac{\delta s}{s}\right)^{2}+\left(\frac{\delta t}{t}\right)^{2}}.
\]
Apply similar rules for \(a\) and for quantities obtained from the SUVAT equations.
Link to Newton’s Second Law
For any straight‑line motion with constant acceleration obtained from a graph:
- Calculate \(a\) from the gradient of the v‑t graph.
- Identify the net force acting (e.g., weight minus tension, or thrust minus drag).
- Check consistency with \(F_{\text{net}} = ma\).
Example: A 2.0 kg cart is pulled horizontally by a constant tension of 10 N and experiences a kinetic‑friction force of 2 N. The measured gradient of the v‑t graph is 4.0 m s\(^{-2}\).
Net force = 10 N − 2 N = 8 N.
Predicted acceleration from Newton’s law: \(a = F/m = 8/2 = 4.0\;\text{m s}^{-2}\), which matches the graphical result, confirming the analysis.
Summary Checklist for Exam Preparation
- Know the definitions and be able to distinguish scalar vs. vector quantities.
- Read straight‑line and curved sections of s‑t, v‑t and a‑t graphs confidently.
- Use a tangent to obtain instantaneous velocity or acceleration.
- Relate average values to the gradient of a secant line and to areas under graphs.
- Derive the five SUVAT equations step‑by‑step; remember which variables each equation contains.
- Apply the graphical method to free fall (with and without air resistance) and to other real‑world scenarios.
- Estimate uncertainties using the half‑grid method and propagate them correctly.
- Connect the measured acceleration to forces via \(F = ma\).