AP Physics B Kinematics Presentation General Problems #11

The New Jersey Center for Teaching and Learning
28 Jun 201211:11
EducationalLearning
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TLDRThis script explains the physics of a baseball hit at an initial velocity of 50 m/s at a 30° angle. It calculates the maximum height reached using the vertical component of the velocity and gravity, determining it to be 32 meters. It also finds the total time in the air to be 5.1 seconds by considering the symmetry of the projectile's motion. The horizontal range is calculated to be 221 meters, using the horizontal component of the velocity and time. Finally, the script determines the velocity of the projectile 5 seconds after launch to be 49 m/s, using the Pythagorean theorem to combine the horizontal and vertical components of velocity.

Takeaways
  • 🏀 The initial speed of the baseball is 50 m/s, launched at a 30° angle above the horizontal.
  • 📉 The vertical acceleration of the projectile is due to gravity, which is -9.8 m/s², indicating a downward force.
  • 🚀 The horizontal acceleration is 0 m/s², as there is no force acting to change the horizontal velocity.
  • 📈 The maximum height formula used is \( \Delta Y = \frac{V_{y_{\text{Apex}}}^2 - V_{y}^2}{2a_y} \), where \( V_{y_{\text{Apex}}} \) is the vertical velocity at the highest point (0 m/s).
  • 🌟 The calculated maximum height of the projectile is 32 meters.
  • ⏱ Two methods are discussed for determining the total time in the air: using the landing equation and the symmetry of the projectile's motion.
  • 🕒 The total time in the air is calculated to be 5.1 seconds using the vertical velocity and acceleration due to gravity.
  • 📏 To find the maximum horizontal distance, the equation \( X = V_x \cdot T \) is used, resulting in a range of 221 meters.
  • 🔢 The velocity components are calculated separately for horizontal (\( V_x \)) and vertical (\( V_y \)) directions.
  • 🛰️ Five seconds after launch, the vertical velocity is -24 m/s, and the horizontal velocity remains 43 m/s due to no horizontal acceleration.
  • 📐 The final velocity 5 seconds after launch is determined using the Pythagorean theorem, resulting in a velocity of 49 m/s.
Q & A

  • What is the initial velocity of the baseball hit in the script?

    -The initial velocity of the baseball is 50 m/s.

  • At what angle is the baseball hit in relation to the horizontal?

    -The baseball is hit at a 30° angle above the horizontal.

  • What is the acceleration due to gravity used in the script for the vertical direction?

    -The acceleration due to gravity used in the script for the vertical direction is 9.8 m/s².

  • Why is the acceleration in the horizontal direction considered to be 0 m/s²?

    -The acceleration in the horizontal direction is considered to be 0 m/s² because there is no force acting upon the object to accelerate it horizontally once it is in motion.

  • What is the formula used to calculate the maximum height reached by the projectile?

    -The formula used to calculate the maximum height is ΔY = (Vy_apex² - Vy²) / (2 * a_y), where Vy_apex is the final vertical velocity at the apex, Vy is the initial vertical velocity, and a_y is the acceleration due to gravity.

  • What is the calculated maximum height reached by the projectile in the script?

    -The calculated maximum height reached by the projectile is 32 meters.

  • How is the total time in the air for the projectile determined in the script?

    -The total time in the air is determined using the equation T = (Vy_final - Vy_initial) / a_y, where Vy_final is the final vertical velocity, Vy_initial is the initial vertical velocity, and a_y is the acceleration due to gravity.

  • What is the total time in the air for the projectile as calculated in the script?

    -The total time in the air for the projectile is calculated to be 5.1 seconds.

  • How is the maximum horizontal distance covered by the projectile calculated?

    -The maximum horizontal distance is calculated using the equation X = VX * T, where VX is the horizontal velocity component and T is the total time in the air.

  • What is the maximum horizontal distance covered by the projectile according to the script?

    -The maximum horizontal distance covered by the projectile is 221 meters.

  • How is the velocity of the projectile 5 seconds after it was fired calculated in the script?

    -The velocity 5 seconds after firing is calculated by determining the vertical (Vy) and horizontal (VX) components of the velocity at that time and then using the Pythagorean theorem to find the resultant velocity (V).

  • What is the velocity of the projectile 5 seconds after it was fired as per the script?

    -The velocity of the projectile 5 seconds after it was fired is 49 m/s.

Outlines
00:00
🚀 Projectile Motion Analysis

This paragraph discusses the physics of a baseball hit at an angle, focusing on the maximum height reached by the projectile. The initial velocity is given as 50 m/s at a 30° angle above the horizontal. The acceleration due to gravity is considered, and the horizontal acceleration is zero due to no external forces. The starting position for both x and y coordinates is set at 0 m. The formula used to calculate the height is derived from the kinematic equation, and the height is found to be 32 meters. The paragraph also explains two methods to determine the total time in the air, opting for the second method which uses the initial and final vertical velocities to find a time of 5.1 seconds.

05:01
⏱ Calculating Time and Horizontal Range

The second paragraph delves into calculating the total time the projectile spends in the air and the maximum horizontal distance it covers. It describes the process of using the kinematic equation to find the time it takes for the object to land, given the initial and final heights are both 0 m. The paragraph then simplifies the calculation by using the initial and final vertical velocities, resulting in a time of 5.1 seconds. To determine the horizontal range, the equation x = vx * T is used, with vx being the horizontal component of the initial velocity and T the total time in the air, yielding a range of 221 meters.

10:01
📏 Velocity Calculation Post-Launch

The final paragraph addresses the calculation of the projectile's velocity 5 seconds after it was launched. It explains the need to determine both the vertical (VY) and horizontal (VX) components of the velocity at this time. The vertical velocity is calculated using the initial vertical velocity, the acceleration due to gravity, and the time elapsed. The horizontal velocity remains constant due to the absence of horizontal acceleration. Using the Pythagorean theorem, the resultant velocity is found by combining the vertical and horizontal components, resulting in a velocity of 49 m/s at the specified time.

Mindmap
Keywords
💡Projectile Motion
Projectile motion is the motion of an object thrown into the air, where it follows a curved path under the influence of gravity. In the video, the theme revolves around analyzing the projectile motion of a baseball hit at an angle, with the focus on calculating its maximum height, time in the air, and horizontal range.
💡Initial Velocity
Initial velocity is the speed at which an object starts moving, including both its magnitude and direction. In the context of the video, the baseball is hit with an initial velocity of 50 m/s at a 30-degree angle, which sets the conditions for the subsequent motion calculations.
💡Angle of Projection
The angle of projection is the angle at which a projectile is launched relative to the horizontal. The video script specifies a 30-degree angle for the baseball, which is critical for determining the vertical and horizontal components of the initial velocity.
💡Acceleration Due to Gravity
Acceleration due to gravity is the acceleration experienced by an object in free fall, directed vertically downward. In the video, it is given as 9.8 m/s² and is used to calculate the vertical motion of the baseball, affecting both the maximum height and the time it takes to reach the ground.
💡Horizontal Component
The horizontal component of velocity is the part of an object's velocity that is in the direction parallel to the ground. The video explains that this component remains constant throughout the motion due to the absence of horizontal acceleration.
💡Vertical Component
The vertical component of velocity is the part of an object's velocity that is in the direction perpendicular to the ground. In the video, it is used to calculate the maximum height reached by the baseball and its time in the air.
💡Apex
The apex of a projectile's trajectory is the highest point it reaches before descending. The video uses the concept of the apex to find the maximum height of the baseball's flight, where the vertical velocity component is zero.
💡Equation of Motion
The equations of motion are formulas used to describe the motion of an object under the influence of forces. The video employs these equations, particularly those involving vertical motion, to calculate the height, time, and range of the projectile.
💡Horizontal Range
Horizontal range is the total horizontal distance covered by a projectile in its flight. The video script calculates the horizontal range of the baseball using the time of flight and the horizontal component of the initial velocity.
💡Time in the Air
Time in the air refers to the total duration a projectile spends from the moment it is launched until it lands. The video script provides two methods to calculate this time, emphasizing the elegance of using the vertical motion equations.
💡Velocity Vector
A velocity vector is a representation of an object's velocity, including both its magnitude and direction. In the video, the velocity vector is decomposed into vertical and horizontal components and then recombined to find the resultant velocity at a specific time after launch.
Highlights

Initial velocity of 50 m/s at a 30° angle above the horizontal for the baseball hit.

Vertical acceleration equals acceleration due to gravity, 9.8 m/s².

Horizontal acceleration is 0 m/s² as there is no force acting on the object.

At the highest point, the vertical component of velocity (Vy) is 0 m/s.

Equation used to calculate height: ΔY = Vy² - Vy₀² / 2ay.

Height of the projectile calculated to be 32 meters.

Two methods to determine total time in the air for the projectile.

Total time in the air calculated to be 5.1 seconds using the second method.

Horizontal range equation: x = vx * T, with no horizontal acceleration.

Maximum horizontal distance covered by the projectile is 221 meters.

Calculating the velocity of the projectile 5 seconds after launch involves determining Vy and vx.

Vy is calculated using the initial vertical velocity component and gravity.

Vx remains constant at 43 m/s due to no horizontal acceleration.

Resultant velocity after 5 seconds calculated using the Pythagorean theorem.

Velocity of the projectile 5 seconds after launch is 49 m/s.

Transcripts

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Projectile MotionPhysics CalculationMathematicsTrajectory AnalysisVertical MotionHorizontal RangeMotion AnalysisSpeed CalculationAcceleration Due to GravityTime in AirVelocity Vector