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MECHANICS FOR PP4

M1 TOPIC 2 : KINEMATICS

CIE A-Level Mechanics 1: Kinematics in a Straight Line

CIE A-Level Mechanics 1

Kinematics of Motion in a Straight Line

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. In this section, we focus on motion in one dimension (straight line).

Note: Restricted to motion in one dimension only. The term 'deceleration' may sometimes be used in the context of decreasing speed.

Scalar and Vector Quantities

Scalar quantities have magnitude only (size).

Vector quantities have both magnitude and direction.

Distance - scalar quantity (m)

Speed - scalar quantity (m/s)

Displacement - vector quantity (m)

Velocity - vector quantity (m/s)

Acceleration - vector quantity (m/s²)

Key Point: The direction of velocity and acceleration vectors is crucial in one-dimensional motion. We use positive and negative signs to indicate direction (e.g., positive for right/up, negative for left/down).

Motion Graphs

Displacement-Time Graphs

Displacement-Time Graph

[Graph: s on y-axis, t on x-axis]

Gradient = Velocity

The gradient (slope) of a displacement-time graph at any point gives the velocity at that instant.

v = ds/dt

Straight line → constant velocity
Curved line → changing velocity (acceleration)
Horizontal line → stationary (zero velocity)
Steeper slope → higher velocity

Velocity-Time Graphs

Velocity-Time Graph

[Graph: v on y-axis, t on x-axis]

Gradient = Acceleration

The gradient (slope) of a velocity-time graph at any point gives the acceleration at that instant.

a = dv/dt

Area under graph = Displacement

The area between the velocity-time graph and the time axis represents the displacement.

s = ∫v dt

Straight line → constant acceleration
Curved line → changing acceleration
Horizontal line → constant velocity (zero acceleration)
Area above time axis → positive displacement
Area below time axis → negative displacement

Calculus in Kinematics

We can use differentiation and integration to solve problems involving displacement, velocity, and acceleration.

v = ds/dt a = dv/dt = d²s/dt²

s = ∫v dt v = ∫a dt

Differentiation (Finding Rates of Change)

Velocity is the rate of change of displacement with respect to time.

Acceleration is the rate of change of velocity with respect to time.

Example: If s = 3t² + 2t + 1, then:

v = ds/dt = 6t + 2

a = dv/dt = 6

Integration (Finding Quantities from Rates)

Displacement is the integral of velocity with respect to time.

Velocity is the integral of acceleration with respect to time.

Example: If a = 4 m/s² and initial velocity u = 2 m/s, then:

v = ∫a dt = ∫4 dt = 4t + C

Using v = 2 when t = 0: 2 = 4(0) + C ⇒ C = 2

v = 4t + 2

Note: Calculus required is restricted to techniques from the content for Paper 1: Pure Mathematics 1.

Motion with Constant Acceleration

When acceleration is constant, we can use the following equations of motion (suvat equations):

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u + v)t

s = displacement (m)

u = initial velocity (m/s)

v = final velocity (m/s)

a = acceleration (m/s²)

t = time (s)

Important: These equations only work when acceleration is constant. Remember to use appropriate signs for direction (+/-).

Problem-Solving Strategy

Write down known quantities (s, u, v, a, t)
Identify the unknown quantity you need to find
Choose the equation that connects known and unknown quantities
Substitute values and solve
Check if your answer makes sense (magnitude and direction)

Note: Questions may involve setting up more than one equation, using information about the motion of different particles.

Equations Summary

Velocity from displacement
v = ds/dt

Acceleration from velocity
a = dv/dt

Displacement from velocity
s = ∫v dt

Velocity from acceleration
v = ∫a dt

Constant acceleration (1)
v = u + at

Constant acceleration (2)
s = ut + ½at²

Constant acceleration (3)
v² = u² + 2as

Constant acceleration (4)
s = ½(u + v)t

Graph Interpretation Summary:

Displacement-time graph: Gradient = Velocity
Velocity-time graph: Gradient = Acceleration, Area = Displacement