Summary Tradisional | Kinematics: Oblique Motion Equation

Contextualization

Kinematics is a branch of physics that explores how objects move without necessarily examining the forces behind that movement. In everyday life, we often see examples of this in projectile motion, like the arc of a tossed ball or the flight path of a rocket. This type of motion happens in two dimensions, which means we can separate the trajectory into horizontal and vertical parts.

Projectile motion is best understood as a mix of two movements: a constant-speed (or uniform) motion in the horizontal direction and a uniformly accelerated motion in the vertical direction, thanks to gravity. Breaking the motion down in this way is key to describing it mathematically and applying our understanding in real-life scenarios, from launching projectiles to analysing sports plays.

To Remember!

Decomposition of Motion

When we look at projectile motion, it can be separated into two parts: the horizontal and the vertical components. The horizontal part features uniform motion, which means the object moves at a constant speed with no acceleration, while the vertical part is subject to uniform acceleration, with gravity altering the speed as time passes.

This separation is fundamental because it lets us apply the correct equations to each component separately. For horizontal movement, we rely on the equations for constant speed, and for vertical movement, we use the formulas for uniformly accelerated motion. This approach is essential for predicting the object’s true path and for calculating key details like its range and maximum height.

• Projectile motion is divided into horizontal and vertical components.

• The horizontal component involves uniform motion, while the vertical component involves uniform acceleration due to gravity.

• This breakdown enables us to apply the right equations to each component individually.

Equations of Uniform and Uniformly Accelerated Motion

To accurately describe projectile motion, we use two main equations. For uniform (horizontal) motion, we use S = S₀ + vt, where S represents the final position, S₀ is the starting position, v is the constant velocity, and t is the time elapsed. This equation works well for the horizontal aspect of projectile motion because the speed remains constant.

For the vertical component, affected by gravity, we apply the formula S = S₀ + vt + 1/2at². Here, S is the final vertical position, S₀ is the initial position, v is the starting velocity, a stands for acceleration (with gravity being approximately 9.8 m/s²), and t is time. Together, these equations allow us to map out the full trajectory of a projectile.

• Uniform motion is described by S = S₀ + vt.

• Uniformly accelerated motion is given by S = S₀ + vt + 1/2at².

• These equations apply to the horizontal and vertical components, respectively.

Maximum Range and Maximum Height

The maximum range of a projectile is the farthest horizontal distance it will travel, and it can be calculated using the formula R = (v₀² * sin(2θ))/g. In this formula, v₀ is the initial velocity, θ is the angle at which the object is launched, and g represents gravitational acceleration. This equation takes into account how the initial velocity is split into horizontal and vertical components.

Similarly, the maximum height refers to the highest point reached during flight. To find this, we use H = (v₀² * sin²(θ))/(2g), which focuses on the vertical component and the effect of gravity. Understanding these calculations is key to solving problems involving projectile motion and has practical applications in fields like engineering and sports.

• The maximum range is found using R = (v₀² * sin(2θ))/g.

• The maximum height is calculated with H = (v₀² * sin²(θ))/(2g).

• These formulas are crucial for practical applications in areas such as engineering and sports.

Problem Solving

Tackling problems in projectile motion involves a few critical steps. First, you need to break down the initial velocity into its horizontal and vertical components using the sine and cosine of the launch angle: v₀x = v₀ * cos(θ) for the horizontal, and v₀y = v₀ * sin(θ) for the vertical.

Next, apply the equations for uniform and uniformly accelerated motion to determine the position and velocity at different times. For the horizontal motion, use S = S₀ + vt; for the vertical motion, apply S = S₀ + vt + 1/2at² and also v = v₀ + at.

Finally, if needed, compute the maximum range and height using the respective formulas. Practicing these steps through guided problem-solving exercises reinforces the concepts and helps deepen your understanding of how projectile motion is analysed.

• Break down the initial velocity into horizontal and vertical components.

• Apply the relevant equations for uniform and accelerated motion.

• Calculate maximum range and height when necessary.

Key Terms

• Kinematics: The study of how objects move without considering why they move.

• Projectile Motion: Motion that occurs in two dimensions and can be split into horizontal and vertical components.

• Uniform Motion: Movement at a constant speed with no acceleration.

• Uniformly Accelerated Motion: Movement where speed changes at a constant rate due to acceleration.

• Decomposition of Motion: The process of separating projectile motion into horizontal and vertical parts.

• Maximum Range: The greatest horizontal distance a projectile can travel.

• Maximum Height: The highest point reached by a projectile during its flight.

• Initial Velocity: The speed at which an object is launched.

• Acceleration due to Gravity: The constant acceleration (approximately 9.8 m/s²) that influences vertical motion.

Important Conclusions

This lesson provided an in-depth look at kinematics, focusing on projectile motion and its division into horizontal and vertical components. We saw that the horizontal motion is uniform while the vertical motion is uniformly accelerated due to gravity. The equations S = S₀ + vt for uniform motion and S = S₀ + vt + 1/2at² for accelerated motion are critical tools for analysing these components.

We also discussed how to calculate key parameters like the maximum range and maximum height using the formulas R = (v₀² * sin(2θ))/g and H = (v₀² * sin²(θ))/(2g). These concepts are not only essential for solving textbook problems but also have practical applications in fields ranging from engineering to sports. Through guided exercises and problem-solving, students are encouraged to build a solid understanding of these principles and explore their real-world applications.

Study Tips

• Review your class notes and practice breaking motions into horizontal and vertical components.

• Work through additional projectile motion problems to get comfortable with the equations for both uniform and accelerated motion.

• Look into real-life examples of projectile motion in sports and engineering to see how these concepts are applied.