Kinematics
2 KinematicsLearners should be able to demonstrate and apply their knowledge and understanding of: |
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| a) | What is meant by displacement, mean and instantaneous values of speed, velocity and acceleration |
| b) | The representation of displacement, speed, velocity and acceleration by graphical methods |
| c) | The properties of displacement-time graphs, velocity-time graphs, and interpret speed and displacement-time graphs for non-uniform acceleration |
| d) | How to derive and use equations which represent uniformly accelerated motion in a straight line |
| e) | How to describe the motion of bodies falling in a gravitational field with and without air resistance – terminal velocity |
| f) | The independence of vertical and horizontal motion of a body moving freely under gravity |
| g) | The explanation of the motion due to a uniform velocity in one direction and uniform acceleration in a perpendicular direction, and perform simple calculations |
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Specified Practical Work o Measurement of g by freefall |
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Learners should be able to demonstrate and apply their knowledge and understanding of:
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a) What is Meant by Displacement, Mean and Instantaneous Values of Speed, Velocity, and Acceleration
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1. Displacement:
- Definition:
- – Displacement is a vector quantity representing the shortest distance and direction between an initial and a final position of an object.
- Difference from Distance:
- – Distance is a scalar, measuring the total path covered, while displacement considers only the direct line between two points.
- Units: Meters (m).
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2. Mean Speed and Instantaneous Speed:
- Mean Speed:
- – Formula:
- [math] \text{Mean Speed} = \frac{\text{Total Distance}}{\text{Total Time}}[/math]
- – Example:
- If an object covers 100 m in 10 s, the mean speed is [math] \frac{100}{10} = 10m/s [/math]
- Instantaneous Speed:
- – Definition: The speed of an object at a specific instant in time.
- – Determined by the slope of the displacement-time graph at a particular point.
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3. Mean Velocity and Instantaneous Velocity:
- Mean Velocity:
- – Formula:
- [math] \text{Mean Velocity} = \frac{\text{Displacement}}{\text{Time}}[/math]
- – Example:
- If an object moves 50 m east and then 30 m west in 10 s, the displacement is 20m, so
- [math]\text{Mean Velocity} = \frac{\text{Displacement}}{\text{Time}} \\
\text{Mean Velocity} = \frac{20}{10} \\
\text{Mean Velocity} = 2 \, \text{m/s} [/math] - Instantaneous Velocity:
- – Definition: The velocity at a specific instant, including direction
- – Determined by the slope of the displacement-time graph at a point.
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4. Mean and Instantaneous Acceleration:
- Mean Acceleration:
- – Formula:
- [math]\text{Mean Acceleration} = \frac{∆v}{∆t}[/math]
- Where ∆v is the change in velocity and ∆t is the time taken.
- – Example:
- If velocity changes from 5 m/s to 15 m/s in 2 s, mean acceleration is:
- [math]a = \frac{15 – 5}{2} \\
a = 5 \, \text{m/s}^2 [/math] - Instantaneous Acceleration:
- – The rate of change of velocity at a specific instant.
- – Determined by the slope of the velocity-time graph at a particular point.
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b) Representation of Displacement, Speed, Velocity, and Acceleration by Graphical Methods
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1. Displacement-Time Graph:
- Slope: Represents velocity (v).
- Straight Line:
- – Positive slope: Constant velocity in a positive direction.
- – Negative slope: Constant velocity in a negative direction.
- Curved Line: Changing velocity (acceleration).
- Figure 1 Displacement-time graph
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2. Velocity-Time Graph:
- Slope: Represents acceleration (a).
- Area Under the Graph: Represents displacement.
- ⇒ Horizontal Line:
- Positive: Constant positive velocity.
- Zero: Object at rest.
- ⇒ Changing Slope: Non-uniform acceleration.
- Figure 2 Velocity-time graph
- 3. Acceleration-Time Graph:
- Horizontal Line:
- – Constant acceleration.
- – Zero: No acceleration (constant velocity).
- Area Under the Graph: Represents change in velocity.
- Figure 3 Acceleration-time graph
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c) Properties of Displacement-Time, Velocity-Time Graphs, and Non-Uniform Acceleration
- Displacement-Time Graphs:
- – Straight-line segments: Uniform velocity.
- – Curved sections: Non-uniform velocity (acceleration or deceleration).
- – Plateau (flat line): Object at rest.
- Velocity-Time Graphs:
- – Positive slope: Positive acceleration.
- – Negative slope: Negative acceleration (deceleration).
- – Curved sections: Non-uniform acceleration.
- Non-Uniform Acceleration:
- – Definition: Acceleration changes over time.
- – Represented by a non-linear velocity-time graph.
- – Calculated using slopes at specific intervals or numerical integration.
- d) Derivation and Use of Equations Representing Uniformly Accelerated Motion
- Uniformly accelerated motion in a straight line is governed by the following equations:
- [math]- v = u + at \\
– s = ut + \frac{1}{2} at^2 \\
– v^2 = u^2 + 2as \\
– s = \frac{(u + v)}{2} \cdot t[/math] - Where:
- – v: Final velocity,
- – u: Initial velocity,
- – a: Acceleration,
- – t: Time,
- – s: Displacement.
- Derivations:
- – From Definition of Acceleration:
- [math]a = \frac{v – u}{t} [/math]
- Rearranging:
- [math]v = u + at[/math]
- From Displacement (s):
- – Average velocity is:
- [math]\text{Average Velocity} = \frac{u + v}{2} [/math]
- – Displacement is:
- [math]s = \text{Average Velocity} \times t = \left( \frac{u + v}{2} \right) t [/math]
- From Substituting
- [math]v = u + at \\ \text{ into } \\ s = ut + \frac{1}{2} at^2:[/math]
- Substituting
- [math]v = u + at [/math]
- into
- [math]v^2 = u^2 + 2as [/math]
- ⇒ Examples of Use:
- Finding Displacement:
- – If u=0 m/s, a=5 m/s2, t=2 s
- [math]s = ut + \frac{1}{2} at^2 \\
s = (0)(2) + \frac{1}{2}(5)(2)^2 \\
s = 0 + \frac{1}{2}(5)(4) \\
s = 10 \, \text{m} [/math] - Finding Final Velocity:
- – If u=0 m/s, a=5 m/s2, t=3s
- [math]v = u + at \\
v = 0 + (5)(3) \\
v = 15 \, \text{m/s} [/math] - – These equations and graphs are crucial for analyzing motion in physics.
- e) Falling Motion in a Gravitational Field Without Air Resistance
- When a body falls freely under gravity in the absence of air resistance:
- – It accelerates uniformly at g=9.81 m/s2 (the acceleration due to gravity near Earth’s surface).
- The motion is described by the equations of uniformly accelerated motion:
- [math]v = u + gt \\
s = ut + \frac{1}{2} gt^2 \\
v^2 = u^2 + 2gs [/math] - Characteristics:
- – Initial velocity (u) is often 0 for a dropped object.
- – Acceleration remains constant at g
- – Velocity increases linearly over time.
- Example:
- – A ball is dropped from rest (u=0) from a height of 20 m. Using [math]s = \frac{1}{2} gt^2[/math]:
- [math]s = \frac{1}{2} gt^2 \\
t = \sqrt{\frac{2s}{g}} \\
t = \sqrt{\frac{2(20)}{9.81}} \\
t = 2.02 \, \text{s}[/math] 
- Figure 4 Free Fall and Air Resistance
- ⇒ Falling Motion in a Gravitational Field With Air Resistance
- Air resistance opposes the motion of a falling object. The force of air resistance increases with velocity until it balances the downward gravitational force.
- Stages of Motion:
- – Initial Acceleration:
- When the object begins to fall, gravitational force ([math]F_g = mg[/math] ) dominates, and the object accelerates downward.
- – Increasing Air Resistance:
- As velocity increases, air resistance increases.
- The net force acting on the object decreases, causing acceleration to reduce.
- – Terminal Velocity:
- When air resistance equals gravitational force ([math]F_g = F_{air}[/math]), the net force becomes zero, and the object stops accelerating.
- The object continues to fall at a constant terminal velocity.
- – Graphical Representation:
- Velocity-time graph: A curve that approaches a horizontal asymptote at the terminal velocity.
- Acceleration-time graph: Starts at g and decreases to zero as terminal velocity is reached.
- Example:
- A skydiver experiences rapid acceleration at first but reaches terminal velocity (around 50 m/s) after a few seconds due to air resistance.
- ⇒ Terminal Velocity
- Definition:
- – The constant velocity reached by a falling object when the force of air resistance equals the force of gravity.
- Factors Affecting Terminal Velocity:
- – Mass and weight of the object
- – Shape and surface area (drag coefficient).
- – Density of the fluid (air or liquid).
- Formula for Terminal Velocity (simplified for a sphere):
- [math]v_t = \sqrt{\frac{2mg}{\rho c_d A}}[/math]
- Where:
- – m: Mass,
- – g: Acceleration due to gravity,
- – ρ: Density of the fluid,
- – [math]c_d[/math]: Drag coefficient,
- – A: Cross-sectional area.
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f) Independence of Vertical and Horizontal Motion Under Gravity
- The vertical and horizontal components of motion are independent of each other when a body moves freely under gravity.
- ⇒ Vertical Motion:
- Governed by gravitational acceleration (g).
- The equations of motion ([math]v = u + gt,[/math] etc) describe vertical motion.
- ⇒ Horizontal Motion:
- With no horizontal forces acting, horizontal velocity ([math]v_x[/math] ) remains constant.
- Horizontal displacement: [math]x = v_x t[/math].
- ⇒ Combined Motion:
- The trajectory of a body moving under gravity is a parabola, resulting from:
- – Uniform velocity in the horizontal direction
- – Uniform acceleration in the vertical direction.
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g) Explanation of the Motion Due to Uniform Velocity in One Direction and Uniform Acceleration Perpendicular
- Projectile Motion:
- – A common example where an object moves with constant velocity in one direction (horizontal) and uniform acceleration in the perpendicular direction (vertical).
- Equations for Projectile Motion:
- Horizontal Motion:
- – [math]x = v_x t[/math]
- – [math]v_x = [/math]constent
- Vertical Motion:
- [math]y = v_{y0} t + \frac{1}{2}(-g)t^2 \\
v_y = v_{y0} – gt[/math] - Trajectory Equation:
- Eliminating t between horizontal and vertical equations gives:
- [math]y = x \tan\theta – \frac{g x^2}{2 v_x^2 \cos^2\theta} [/math]
- This represents the parabolic path of the projectile.
- ⇒ Example Calculations
- ⇒ Projectile Launched at an Angle:
- A ball is launched with an initial velocity[math]v_0 = 20 \, \text{m/s}, \quad \theta = 30^\circ[/math].
- ⇒ Horizontal and Vertical Components:
- [math]v_x = v_0 \cos\theta \\
v_x = 20 \cos 30^\circ \\
v_x = 17.32 \, \text{m/s} \\
v_y = v_0 \sin\theta \\
v_y = 20 \sin 30^\circ \\
v_y = 10 \, \text{m/s}[/math] - Time of Flight:
- Using
- [math]y = v_{y0} t – \frac{1}{2} g t^2 = 0[/math]
- (at landing):
- [math]t = \frac{2v_y}{g} \\
t = \frac{2(10)}{9.81} \\
t = 2.04 \, \text{s}[/math] - Horizontal Range:
- Using
- [math]x = v_x t \\
x = (17.32)(2.04) \\
x = 35.3 \, \text{m}[/math] - ⇒ Dropping Objects with Air Resistance:
- An object with mass 5 kg, cross-sectional area 05 m2, and drag coefficient 0.5:
- Estimate using the terminal velocity formula.
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Specified Practical Work
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Measurement of g (Acceleration Due to Gravity) Using Freefall
- The experiment to measure g involves observing the motion of a freely falling object and determining the time it takes to fall a known distance.
- Various methods can be used, but one common and precise method uses an electromagnetic release mechanism and a trapdoor system or light gates.
- Objective
- To determine the acceleration due to gravity (g) by measuring the time for an object to fall a known vertical distance.
- Equipment
- – Electromagnet or release mechanism.
- – Steel or brass ball (or another dense, symmetrical object).
- – Trapdoor mechanism or light gate.
- – Millisecond timer or data logger.
- – Meter ruler or another accurate measuring device for the height.
- – Clamp stand to hold the setup in place.
- Figure 5 Measurement of g (Acceleration Due to Gravity) Using Freefall
- Theory
- The object falls freely under gravity. For freefall, the distance s and time t are related by the equation of uniformly accelerated motion:
- [math]s = \frac{1}{2} g t^2[/math]
- Rearranging for g:
- [math]g = \frac{2s}{t^2}[/math]
- By measuring s (distance) and t (time), g can be calculated.
- ⇒ Experimental Setup
- Attach the ball to an electromagnet to hold it at a height s.
- Set up a trapdoor or light gate at the bottom to measure when the ball reaches the ground.
- – Trapdoor method: When the ball hits the trapdoor, it completes an electrical circuit, stopping the timer.
- – Light gate method: The ball passes through the light gate, which stops the timer.
- Measure the vertical height s from the bottom of the ball (at rest position) to the trapdoor or light gate using a meter ruler.
- ⇒ Procedure
- 1. Setup:
- Secure the electromagnet and trapdoor/light gate firmly to avoid misalignment during the experiment.
- Set the timer to zero.
- 2. Release:
- Release the ball by deactivating the electromagnet. This ensures that the ball starts falling freely without external forces.
- 3. Measurement:
- Record the time t for the ball to fall and the corresponding height s.
- 4. Repeat:
- Repeat the experiment for various heights (e.g., 0.5 m) to obtain multiple readings
- Perform at least 3 trials for each height to reduce random errors.
- ⇒ Data Analysis
- – Record the measured distances (s) and times (t) in a table.
- – Plot a graph of s against [math]t^2[/math].
- The equation
- [math]s = \frac{1}{2} g t^2 [/math]
- Suggests that the slope of the graph is [math]\frac{1}{2}g[/math].
- Calculate g from the slope as:
- [math]g = 2 × slope [/math]
- ⇒ Sources of Error
- 1. Reaction Time: Human error in starting/stopping the timer (mitigated using automated timing systems).
- 2. Air Resistance: May slightly reduce the acceleration of the ball.
- 3. Measurement Error: Errors in measuring the height s or timing t.
- 4. Alignment Issues: Misalignment of the ball with the trapdoor/light gate can cause inaccuracies.
- ⇒ Improvements
- 1. Use light gates connected to a data logger for more precise timing.
- 2. Ensure the ball’s path is perfectly vertical to avoid horizontal displacement.
- 3. Repeat the experiment multiple times to calculate an average value of g.
- ⇒ Example Calculation
- Suppose the following data is collected:
| s(m) | t (s) | [math]t^2 (s^2)[/math] |
|---|---|---|
| 0.5 | 0.319 | 0.102 |
| 1.0 | 0.452 | 0.204 |
| 1.5 | 0.553 | 0.306 |
- From the graph of s vs.[math]t^2[/math] :
- Slope = 91 m/s2
- Calculated
- [math]g = 2 \times \text{slope} \\
g = 2 \times 4.91 \\
g = 9.82 \, \text{m/s}^2 [/math] - ⇒ Conclusion
- The experiment provides a reliable measurement of g, close to the accepted value ( [math]9.82 \, \text{m/s}^2 [/math]), depending on experimental conditions and precision of the apparatus.