Chapter 8: 2D Dynamics – Kinematics, Dynamics, and Projectile Motion with Drag

Chapter 8: 2D Dynamics

Introduction to 2D Dynamics

Two-dimensional dynamics involves analyzing the motion of objects in a plane, considering both kinematics (description of motion) and dynamics (forces causing motion). This chapter builds on foundational concepts from previous chapters, focusing on vector equations and their components.

• Kinematics describes how position, velocity, and acceleration change over time.

• Dynamics explains how forces produce acceleration according to Newton's Second Law.

Key Equations:

• Kinematics: $\vec{v}(t) = \frac{d\vec{r}(t)}{dt}$ and $\vec{a}(t) = \frac{d\vec{v}(t)}{dt}$

• Dynamics: $\vec{F} = m\vec{a}$

Component Equations in 2D Motion

Vector and Component Representation

Vectors in two dimensions are expressed in terms of their x and y components. This allows us to analyze motion and forces separately along each axis.

• Position Vector: $\vec{r} = x\hat{i} + y\hat{j}$

• Velocity Vector: $\vec{v} = v_x\hat{i} + v_y\hat{j}$

• Acceleration Vector: $\vec{a} = a_x\hat{i} + a_y\hat{j}$

• Force Vector: $\vec{F} = F_x\hat{i} + F_y\hat{j}$

Component Equations:

Equation	x-component	y-component
Velocity	$v_x = \frac{dx}{dt}$	$v_y = \frac{dy}{dt}$
Acceleration	$a_x = \frac{dv_x}{dt}$	$a_y = \frac{dv_y}{dt}$
Newton's 2nd Law	$F_x = m a_x$	$F_y = m a_y$

Constant Acceleration Equations

For constant acceleration, the kinematic equations can be integrated to yield:

• $x_f = x_i + v_{x,i} \Delta t + \frac{1}{2} a_x (\Delta t)^2$

• $v_{x,f} = v_{x,i} + a_x \Delta t$

• $v_{x,f}^2 = v_{x,i}^2 + 2 a_x \Delta x$

• $\Delta x = x_f - x_i$

• $F_x = m a_x$

• $y_f = y_i + v_{y,i} \Delta t + \frac{1}{2} a_y (\Delta t)^2$

• $v_{y,f} = v_{y,i} + a_y \Delta t$

• $v_{y,f}^2 = v_{y,i}^2 + 2 a_y \Delta y$

• $\Delta y = y_f - y_i$

• $F_y = m a_y$

Example: Calculating the final position and velocity of a projectile under constant acceleration.

Projectile Motion and Aerodynamic Drag

Projectile Motion without Drag

In the absence of air resistance, a projectile follows a parabolic trajectory determined by its initial velocity and the acceleration due to gravity.

• Maximum range is achieved when the launch angle is 45° if the projectile lands at the same elevation from which it was launched.

Aerodynamic Drag Model

When air resistance is considered, the drag force opposes the motion and depends on the velocity squared. The drag force is given by:

• $\vec{D} = \left( \frac{1}{2} \rho C A v^2, \text{direction} = -\vec{v} \right)$

• A: Cross-sectional area

• ρ: Air density

• C: Drag coefficient

• m: Mass

The acceleration components due to gravity and drag are:

• $a_x = -\frac{\rho C A}{2m} v_x \sqrt{v_x^2 + v_y^2}$

• $a_y = -g - \frac{\rho C A}{2m} v_y \sqrt{v_x^2 + v_y^2}$

Note: These are coupled equations; $a_x$ depends on $v_x$ and $v_y$, and $a_y$ depends on $v_x$ and $v_y$.

Examples of Trajectories with Air Drag

Numerical simulations (such as PhET) are used to solve the coupled equations for projectile motion with drag. Real-world examples include golf ball trajectories, which are significantly affected by air resistance and spin.

• Golf ball trajectories: Different clubs produce different ranges due to varying initial velocities and angles.

• Spin effects: The spin of the ball and the presence of dimples can create additional lift and sideways forces, altering the trajectory.

• Other sports: Baseballs and soccer balls also exhibit curving trajectories due to aerodynamic effects.

Whiteboard Problems

Whiteboard Problem 8-1: Rocket-Powered Hockey Puck

A rocket-powered hockey puck with a thrust of 2.0 N and mass 1.0 kg is released from rest on a frictionless table, 4.0 m from the edge of a 2.0 m drop. Neglecting aerodynamic drag, the problem asks how far the puck lands from the base of the table.

• Assumption: The puck does not rotate, so the thrust remains horizontal.

• Application: Use constant acceleration equations to solve for the horizontal and vertical motion.

Whiteboard Problem 8-2: Kicking into the Wind

With a constant horizontal drag force (headwind), the range and optimal launch angle of a projectile are affected. The drag force is $\vec{F}_w = -F_w \hat{i}$.

• Example: Field goal kicked into the wind.

• Task: Find expressions for the range and the angle for maximum range.

Projectile Motion with Drag – Table

Parameter	Without Drag	With Drag
Trajectory Shape	Parabolic	Non-parabolic, shorter range
Maximum Range Angle	45°	Less than 45° (depends on drag)
Equations of Motion	Decoupled, simple	Coupled, require numerical solution
Key Forces	Gravity	Gravity, Drag

Additional Info

• PhET simulations are useful for visualizing projectile motion with and without drag.

• Spin and surface features (like dimples) can significantly affect the trajectory of sports balls.