4.3 Projectile Motion

Projectile motion combines horizontal and vertical movements, creating parabolic trajectories. It's crucial in understanding how objects move through the air, from baseballs to rockets. This concept ties together key physics principles like velocity, acceleration, and gravity.

Analyzing projectile motion involves breaking it down into separate horizontal and vertical components. This approach allows us to calculate important parameters like range, time of flight, and maximum height. These calculations help predict and control object trajectories in real-world applications.

Projectile Motion

Analysis of projectile motion

• Projectile motion combines two independent one-dimensional motions
  • Horizontal motion (x-direction) has constant velocity ($v_x$) and no acceleration ($a_x = 0$)
  • Vertical motion (y-direction) has constant acceleration due to gravity ($a_y = -g$) and initial vertical velocity ($v_{0y}$) depends on launch angle ($\theta$)
• Equations of motion for each direction describe the projectile's position over time
  • Horizontal equation: $x = v_{0x}t$ where $v_{0x}$ is the initial horizontal velocity
  • Vertical equation: $y = v_{0y}t - \frac{1}{2}gt^2$ accounts for gravity's effect on vertical position
• The acceleration due to gravity (gravitational acceleration) is typically denoted as $g$ and is approximately 9.8 m/s² near Earth's surface

Parameters of flat surface projectiles

• Range ($R$) is the horizontal distance traveled by the projectile before landing, calculated using $R = \frac{v_0^2\sin(2\theta)}{g}$
  • Maximum range is achieved with a 45° launch angle (neglecting air resistance)
• Time of flight ($t_f$) represents the total time the projectile is airborne, found with $t_f = \frac{2v_{0y}}{g} = \frac{2v_0\sin(\theta)}{g}$
  • Time of flight depends on initial vertical velocity and gravity
• Maximum height ($h_{max}$) is the highest vertical position reached at half the time of flight ($\frac{t_f}{2}$), determined by $h_{max} = \frac{v_{0y}^2}{2g} = \frac{v_0^2\sin^2(\theta)}{2g}$
  • Maximum height is influenced by initial velocity and launch angle

Projectiles with height differences

• Time of flight for projectiles landing at different heights is found using the vertical motion equation $y = v_{0y}t - \frac{1}{2}gt^2$ and solving for $t$ with the quadratic formula
  • Height difference affects time of flight and trajectory shape
• Impact velocity ($v_f$) has horizontal and vertical components
  • Horizontal component ($v_{fx}$) equals initial horizontal velocity ($v_{0x}$)
  • Vertical component ($v_{fy}$) depends on time of flight and initial vertical velocity: $v_{fy} = v_{0y} - gt_f$
  • Resultant impact velocity is calculated using $v_f = \sqrt{v_{fx}^2 + v_{fy}^2}$

Trajectories under initial conditions

• Projectile trajectory follows a parabolic path due to combined horizontal and vertical motion
  • Factors influencing trajectory include initial velocity ($v_0$), launch angle ($\theta$), and height difference between launch and landing points
• Trajectory equation is derived by eliminating time ($t$) from horizontal and vertical equations of motion
  1. Horizontal equation: $x = v_{0x}t$
  2. Vertical equation: $y = v_{0y}t - \frac{1}{2}gt^2$
  3. Resulting trajectory equation: $y = \tan(\theta)x - \frac{gx^2}{2v_0^2\cos^2(\theta)}$
• Manipulating initial conditions (velocity, angle, height) allows for precise targeting and trajectory control in applications like sports (basketball) and ballistics (artillery)

Real-world considerations

• Free fall is a special case of projectile motion where an object is subjected only to gravitational force, with no initial horizontal velocity
• Air resistance introduces a drag force that opposes the motion of the projectile, affecting its trajectory
• Terminal velocity is reached when the drag force equals the gravitational force, resulting in constant velocity for a falling object