Projectile Motion: Horizontal Launches

Unit Overview: Two-Dimensional Kinematics (Projectile Motion)

• This unit is relatively short, approximately two and a half weeks long.

• It builds upon concepts from one-dimensional kinematics, applying them to two dimensions.

• Familiarity with one-dimensional free fall is a foundational prerequisite for this unit.

Definition of Free Fall and Acceleration Due to Gravity

• Free Fall: Assumes the absence of air resistance.

• Acceleration due to gravity ($g$):

• Its value is approximately $9.8 \text{ m/s}^2$, (sometimes referenced as $9.82 \text{ m/s}^2$).

• The direction of this acceleration is always downwards.

• Force of Gravity: In free fall, gravity is the only force acting on an object.

• Vector Nature: Since gravity is a vertical force, and it is the only force, all objects in free fall accelerate only in the vertical direction (downwards).

• Unbalanced Forces: Unbalanced forces cause acceleration. In projectile motion, the unbalanced force is gravity, leading to a downward acceleration vector.

Independence of Horizontal and Vertical Motion

• This is a fundamental principle: The horizontal and vertical components of projectile motion are entirely independent of each other.

• One component's behavior does not affect the other's.

• This means horizontal movement does not influence the time it takes to fall vertically, and vertical movement does not influence horizontal distance covered given horizontal velocity.

Horizontal Motion Characteristics

• Horizontal Acceleration ($a<em>x$): There is no horizontal acceleration ($a</em>x = 0$) because gravity acts only vertically. Free fall assumes no other horizontal forces.

• Horizontal Velocity ($v<em>x$): Consequently, the horizontal velocity ($v</em>x$) of a projectile remains constant throughout its flight.

• Kinematic Equation for Horizontal Motion: Due to $a_x = 0$, only one kinematic equation is useful for horizontal motion:

• $d<em>x = v</em>{ix} \cdot t$

• Where $d<em>x$ is horizontal distance, $v</em>{ix}$ is initial horizontal velocity (which is constant), and $t$ is time.

• Other kinematic equations simplify:

  • $v<em>{fx}^2 = v</em>{ix}^2 + 2a<em>xd</em>x \implies v<em>{fx}^2 = v</em>{ix}^2 \implies v<em>{fx} = v</em>{ix}$

  • $d<em>x = v</em>{ix}t + \frac{1}{2}a<em>xt^2 \implies d</em>x = v_{ix}t$

  • $v<em>{fx} = v</em>{ix} + a<em>xt \implies v</em>{fx} = v_{ix}$

Vertical Motion Characteristics (for Horizontal Launches)

• Vertical Acceleration ($a_y$): The vertical acceleration is always $g = 9.8 \text{ m/s}^2$ downwards.

• Initial Vertical Velocity ($v<em>{iy}$): For an object launched perfectly horizontally, its initial vertical velocity ($v</em>{iy}$) is zero.

• Changing Vertical Velocity: Because there is a constant downward acceleration, the vertical velocity of the object constantly changes (increases in the downward direction) over time. This is visually represented by a shrinking upward vertical velocity arrow as the object rises, and a growing downward vertical velocity arrow as it falls.

• Vertex (Highest Point): At the object's highest point (vertex), the vertical velocity is instantaneously zero. However, the object's overall velocity is not zero because it still possesses horizontal velocity.

• Graphs for Vertical Motion:

• Vertical Position-Time Graph: A parabola (indicating a quadratic function).

• Vertical Velocity-Time Graph: A linear graph with a slope of $-9.8 \text{ m/s}^2$ (if upward is defined as positive).

• Kinematic Equation for Vertical Motion (Example - Finding Time for a Drop):

• Given height ($h$), initial vertical velocity ($v<em>{iy} = 0$ for a drop or horizontal launch), and vertical acceleration ($a</em>y = g$),

• The relevant equation is $h = v<em>{iy}t + \frac{1}{2}a</em>yt^2$.

• Substituting $v<em>{iy} = 0$ and $a</em>y = g$ (or $-g$ if upward is positive and $h$ is displacement),

• $h = \frac{1}{2}gt^2$

• Solving for time ($t$): $t = \sqrt{\frac{2h}{g}}$

Connecting Horizontal and Vertical Motion: The Role of Time

• While horizontal and vertical components are independent, the time ($t$) over which the motion occurs is the same for both components.

• The duration an object is in the air vertically is identical to the duration it travels horizontally.

• Demonstration: An object dropped straight down from a certain height and an object launched horizontally from the same height at the same instant will hit the ground at the same time.

• This holds true regardless of the initial horizontal speed (e.g., throwing a ball vs. shooting a bullet horizontally).

• The horizontal speed does not influence the time it takes for the object to fall vertically due to gravity.

• Example Calculation: If time ($t$) is found from vertical motion (e.g., using $t = \sqrt{\frac{2h}{g}}$), this same time can then be used in the horizontal motion equation ($d<em>x = v</em>{ix} \cdot t$) to find the horizontal distance traveled.

Visualizing Projectile Motion

• Dot Diagram (Falling Object): Dots appear farther apart as the object falls, indicating increasing speed.

• Vector Diagram (Projectile):

• The horizontal velocity arrow remains constant in length.

• The vertical velocity arrow grows longer in the downward direction, starting from zero (for horizontal launch).

• The acceleration vector (yellow arrow in demonstration) always points straight downwards, representing gravity.

• When the acceleration is in the opposite direction of velocity (e.g., vertical velocity going up, acceleration pulling down), the object slows down. This is seen as the vertical velocity arrow shrinking as it approaches the highest point.

Applying Kinematic Equations in Two Dimensions

• By understanding that horizontal acceleration is zero and vertical acceleration is $g$ (downwards), the four standard kinematic equations from 1D motion can be applied separately to the horizontal and vertical components, with time ($t$) linking them.

• This approach helps avoid memorizing many new equations specific to 2D projectile motion.

• Variables for a horizontal launch:

  • Horizontal: $a<em>x = 0$, $v</em>{ix}$ (constant), $d_x$, $t$

  • Vertical: $a<em>y = g$, $v</em>{iy} = 0$, $d<em>y$ (or $h$), $v</em>{fy}$, $t$

• Final velocity upon hitting the ground involves combining the constant horizontal velocity ($v<em>{fx} = v</em>{ix}$) and the calculated final vertical velocity ($v_{fy}$) vectorially.