Linear Kinematics in Two Dimensions

Linear Kinematics in Two Dimensions & Projectile Motion

• Two-dimensional kinematics involves movement in both x and y directions, analyzing how objects move in a plane rather than just along a single line.

• Trajectory of projectile motion is called a parabola, which is a symmetrical curve influenced by gravity. Understanding this parabolic path is essential for predicting the range and height of projectiles.

• Factors affecting horizontal range: initial velocity and takeoff angle. The range is maximized when the takeoff angle is 45 degrees, assuming no air resistance.

Projectile Motion

• Airborne body subjected only to gravity (and sometimes negligible air resistance). This means the only force acting upon the object is the Earth's gravitational pull.

• Trajectory (path of a projectile) is called a parabola. This path is predictable and can be calculated using kinematic equations.

• Examples: sports activities like somersaults, baseball, long jump, archery, and basketball. These examples help illustrate the principles of projectile motion in real-world scenarios.

• In outer space (zero gravity), trajectory would be a straight line because there is no gravitational force to curve the path.

Equations of Motion

• X-direction: horizontal, positive to the right. This convention helps in setting up the problem and interpreting the results.

• Y-direction: vertical, positive upward. This is the standard convention used in physics for analyzing vertical motion.

• H model is applied separately to x and y directions to simplify the analysis. This allows us to treat the horizontal and vertical components of motion independently.

• Final position in x-direction: $x<em>1 = x</em>0 + \Delta x$

• Displacement in x-direction: $\Delta x = v<em>{x0} \Delta t + \frac{1}{2} a</em>x (\Delta t)^2$

• Change of velocity in x-direction: $\Delta v<em>x = a</em>x \Delta t$

• Similar equations apply to the y-direction, replacing x-components with y-components to analyze vertical motion.

Acceleration Due to Gravity

• Vertical direction acceleration is constant: $a_y = -9.81 \frac{m}{s^2}$. This value is crucial for calculating vertical motion parameters.

• Acts downward (negative sign). This indicates the direction of the gravitational force.

• Magnitude: $g = ||a_y|| = 9.81 \frac{m}{s^2}$.

Projectile Motion Along Y Direction Equations

• Vertical direction acceleration: $a_y = -9.81 \frac{m}{s^2}$. Constant acceleration due to gravity affects vertical motion.

• Change of velocity: $\Delta v<em>y = a</em>y \Delta t$

• Vertical direction displacement: $\Delta y = v*{y0} \Delta t + \frac{1}{2} a_y (\Delta t)^2$

Deriving Projectile Motion Equations

• $\Delta v_y$ is the area covered by the acceleration and time interval on a velocity-time graph.

• The slope of the velocity curve is the acceleration.

• With constant negative acceleration, the velocity-time curve is a straight line going down, showing a consistent decrease in upward velocity.

• $\Delta y$ is the area covered by the velocity-time curve.

• $\Delta y = v*{y0} \Delta t + \frac{1}{2} a_y (\Delta t)^2$

Trajectory of Vertical Motion

• Acceleration: constant and negative due to gravity.

• Velocity: Increases linearly downward due to constant acceleration.

• Vertical Direction displacement. Delta y is a non linear one, showing a curved path influenced by gravity.

Gravity Effects on Vertical Direction

• Gravity alters projectile path on the Y direction, causing it to curve downwards.

• Magnitude is not constant due to increasing of order item, indicating the cumulative effect of gravity over time.

• The relation between delta y and delta t is not linear because of order item, resulting in a parabolic trajectory.

Horizontal Motion Equations

• No gravity in the horizontal direction: $a_x = 0$ if we neglect air resistance.

• Change of velocity in the horizontal direction is always zero, meaning the horizontal velocity remains constant throughout the motion.

• Velocity in the horizontal direction is constant without external forces.

• Displacement along the horizontal direction increases linearly, as the velocity is constant.

Combining Vertical and Horizontal Motion

• Instantaneous velocity can be decomposed into x and y components, allowing for separate analysis of motion in each direction.

• Horizontal velocity is not affected by gravity (constant), assuming negligible air resistance.

• Vertical velocity is affected by gravity (changes), decreasing as the object moves upward and increasing as it falls.

• Vertical displacement is affected by gravity (changes), resulting in a curved trajectory.

Simple Cases: Free Fall

• Movement only along the y-direction, so the initial velocity is zero at y direction, simplifying the kinematic equations.

• Displacement: Use equations to solve this displacement with the information provided, applying the appropriate kinematic formulas.

• Falling Duration