Cambridge A-Level Mathematics 9709

Mechanics (M2)

Kinetics of motion in 2 dimensions: Displacement, Velocity, Acceleration, Projectile Motion

This section covers the fundamental concepts of kinematics applied to motion in two dimensions. We will explore displacement, velocity, acceleration, and the motion of projectiles under constant acceleration.

1. Displacement, Velocity, and Acceleration

These are the core kinematic quantities. Understanding their definitions and relationships is crucial.

1.1 Displacement

Displacement is the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction.

Mathematically, if an object moves from position $r_i$ to position $r_f$ in a straight line, the displacement $r$ is given by:

$$ \mathbf{r} = r_f - r_i $$

The magnitude of the displacement is the straight-line distance between the initial and final positions.

1.2 Velocity

Velocity is the rate of change of displacement. It is also a vector quantity.

The average velocity is defined as:

$$ \mathbf{v}_{avg} = \frac{\Delta \mathbf{r}}{\Delta t} = \frac{\mathbf{r} - \mathbf{r}_0}{t - t_0} $$

The instantaneous velocity is the velocity at a specific point in time.

The magnitude of the velocity is the speed, which is a scalar quantity.

1.3 Acceleration

Acceleration is the rate of change of velocity. It is also a vector quantity.

The average acceleration is defined as:

$$ \mathbf{a}_{avg} = \frac{\Delta \mathbf{v}}{\Delta t} = \frac{\mathbf{v} - \mathbf{v}_0}{t - t_0} $$

The instantaneous acceleration is the acceleration at a specific point in time.

2. Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity.

2.1 Horizontal and Vertical Motion

In projectile motion, the horizontal and vertical components of motion are independent of each other (assuming no air resistance). The horizontal motion is at a constant velocity, and the vertical motion is at a uniformly accelerated rate due to gravity.

2.2 Equations of Motion

We can analyze the horizontal and vertical motion separately using the standard equations of motion:

• Horizontal motion: $x = v_x t$ (constant velocity)
• Vertical motion: $y = v_{y0} t - \frac{1}{2} g t^2$ (with initial vertical velocity $v_{y0}$ and acceleration due to gravity $g$)
• Vertical velocity: $v_y = v_{y0} - g t$
• Vertical velocity squared: $v_y^2 = v_{y0}^2 - 2 g (y - y_0)$

2.3 Range, Maximum Height, and Time of Flight

We can calculate key parameters of projectile motion:

• Range (R): The horizontal distance traveled by the projectile. $$ R = \frac{v_{0x} (v_{0x} + v_{0y})}{g} $$
• Maximum Height (H): The highest vertical position reached by the projectile. $$ H = \frac{v_{0y}^2}{2g} $$
• Time of Flight (T): The total time the projectile is in the air. $$ T = \frac{2 v_{0y}}{g} $$

2.4 Angle of Projection

The maximum range of a projectile is achieved when the angle of projection is 45 degrees (assuming level ground). The range for a given initial speed is maximized at this angle.

3. Numerical Examples

Consider a projectile launched with an initial speed $v_0$ at an angle $\theta$ above the horizontal. We can analyze its motion by resolving the initial velocity into horizontal and vertical components:

Initial horizontal velocity: $v_{0x} = v_0 \cos \theta$

Initial vertical velocity: $v_{0y} = v_0 \sin \theta$

Using these components, we can determine the horizontal and vertical positions and velocities of the projectile at any time $t$.

Quantity	Formula	Units
Displacement	$\mathbf{r} = r_f - r_i$	m
Average Velocity	$\mathbf{v}_{avg} = \frac{\Delta \mathbf{r}}{\Delta t}$	m/s
Average Acceleration	$\mathbf{a}_{avg} = \frac{\Delta \mathbf{v}}{\Delta t}$	m/s$^2$
Range (R)	$R = \frac{v_{0x} (v_{0x} + v_{0y})}{g}$	m
Maximum Height (H)	$H = \frac{v_{0y}^2}{2g}$	m
Time of Flight (T)	$T = \frac{2 v_{0y}}{g}$	s