AP Physics B Kinematics Presentation General Problems #12

The New Jersey Center for Teaching and Learning
28 Jun 201212:06
EducationalLearning
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TLDRThis script explains the physics of a projectile launched from a 95-meter cliff with an initial velocity of 15 m/s at a 37-degree angle. It calculates the maximum height reached at 141 meters, the total time in the air as 8.4 seconds, and the horizontal distance covered at 335 meters. The script also determines the velocity of the projectile just before impact at the cliff's base, which is 66 m/s, using fundamental projectile motion equations and the Pythagorean theorem.

Takeaways
  • ๐ŸŽฏ The initial speed of the projectile is 50 meters per second at an angle of 37 degrees above the horizontal.
  • ๐Ÿงฎ The vertical acceleration is -9.8 meters per second squared and the horizontal acceleration is 0 meters per second squared.
  • โ›ฐ๏ธ The cliff from which the projectile is fired is 95 meters high.
  • ๐Ÿš€ The maximum height reached by the projectile above the ground is calculated to be 141 meters.
  • โฑ๏ธ The total time the projectile is in the air is determined to be 8.4 seconds.
  • ๐Ÿ“ The maximum horizontal distance covered by the projectile is 335 meters.
  • ๐Ÿ“ To find the velocity just before the projectile hits the ground, the vertical and horizontal components of the velocity are calculated.
  • ๐Ÿ”ฝ The vertical component of the velocity just before impact is -52 meters per second.
  • โžก๏ธ The horizontal component of the velocity just before impact is 40 meters per second.
  • ๐Ÿ“Š The resultant velocity of the projectile just before it hits the ground is 66 meters per second.
Q & A
  • What is the initial speed of the projectile fired from the cliff?

    -The initial speed of the projectile is 15 meters per second.

  • What is the height of the cliff from which the projectile is launched?

    -The cliff is 95 meters high.

  • What is the angle at which the projectile is launched above the horizontal?

    -The projectile is launched at an angle of 37 degrees above the horizontal.

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

    -The formula used is \( y_{\text{apex}}^2 = v_{0y}^2 + 2a_{y}\Delta y \).

  • What is the vertical acceleration acting on the projectile?

    -The vertical acceleration is \( g \), which is -9.8 meters per second squared.

  • How is the maximum height calculated in the script?

    -The maximum height is calculated by rearranging the formula to solve for \( y \) and plugging in the given values.

  • What is the total time the projectile spends in the air before hitting the ground?

    -The total time in the air is 8.4 seconds.

  • What formula is used to determine the time of flight of the projectile?

    -The formula used is \( y = y_0 + v_{0y}T + \frac{1}{2}a_yT^2 \), which is rearranged to form a quadratic equation in terms of \( T \).

  • What is the maximum horizontal distance covered by the projectile?

    -The maximum horizontal distance covered is 335 meters.

  • How is the horizontal distance calculated?

    -The horizontal distance is calculated using the formula \( x = x_0 + v_{0x}T \), with \( x_0 = 0 \) and \( a_x = 0 \).

  • What is the velocity of the projectile just before it hits the bottom of the cliff?

    -The velocity of the projectile just before impact is 66 meters per second.

  • How is the final velocity of the projectile calculated?

    -The final velocity is calculated using the Pythagorean theorem, by finding the square root of the sum of the squares of the horizontal and vertical velocity components.

Outlines
00:00
๐Ÿš€ Projectile Motion Analysis

This paragraph discusses the physics of projectile motion, specifically the calculation of the maximum height reached by a projectile. The given scenario involves a projectile launched from a 95-meter cliff with an initial velocity of 15 m/s at a 37-degree angle. The vertical acceleration due to gravity is -9.8 m/sยฒ, and the horizontal acceleration is 0. The initial vertical position is 95 meters, and the horizontal position is assumed to be zero. The paragraph explains the use of the formula for the maximum height (y_apex) in projectile motion, which is derived from the kinematic equations, to find that the projectile reaches a height of 141 meters above the ground.

05:03
โฑ Calculating Time of Flight

The second paragraph focuses on determining the total time the projectile spends in the air. It starts by setting up the kinematic equation for vertical motion, considering the initial vertical position (95 meters) and the final vertical position (0 meters at ground level). The paragraph then transforms the equation into a quadratic form to solve for the time of flight (T). Using the quadratic formula, the paragraph calculates the time it takes for the projectile to reach the bottom of the cliff, which is found to be 8.4 seconds, disregarding the negative solution as it is not physically meaningful in this context.

10:04
๐Ÿ“ Determining Horizontal Range

In this paragraph, the calculation of the horizontal distance covered by the projectile is presented. By using the horizontal component of the initial velocity and the time of flight determined in the previous paragraph, the horizontal range is calculated. Since there is no horizontal acceleration, the horizontal motion is uniform, and the range is found by multiplying the horizontal component of the initial velocity (50 m/s at a 37-degree angle) by the total time of flight (8.4 seconds), resulting in a horizontal distance of 335 meters.

๐ŸŒช Final Velocity Components

The final paragraph addresses the determination of the projectile's velocity just before it hits the bottom of the cliff. It breaks down the velocity into its vertical (V_y) and horizontal (V_x) components. The vertical component is calculated using the initial vertical velocity, the acceleration due to gravity, and the time of flight, resulting in a final vertical velocity of -52 m/s. The horizontal component remains constant due to zero horizontal acceleration, resulting in a horizontal velocity of 40 m/s. The paragraph then combines these components to find the resultant velocity at impact using the Pythagorean theorem, yielding a velocity of 66 m/s.

Mindmap
Keywords
๐Ÿ’กProjectile
A projectile is an object that is given an initial velocity in a particular direction and then continues in motion due to its inertia unless acted upon by external forces. In the video, a projectile is fired from the edge of a cliff, and the concept is central to understanding the physics of its motion and the calculations involved.
๐Ÿ’กInitial Velocity
Initial velocity is the speed at which an object starts moving. In the context of the video, the projectile is launched with an initial velocity of 15 meters per second at an angle, which is crucial for determining the trajectory and range of the projectile.
๐Ÿ’กAngle of Projection
The angle of projection refers to the angle at which a projectile is launched relative to the horizontal. In the video, the projectile is fired at a 37-degree angle above the horizontal, which affects both the maximum height reached and the horizontal distance covered.
๐Ÿ’กVertical Acceleration
Vertical acceleration in this context is the acceleration due to gravity, which is approximately -9.8 m/s^2. It acts downward and affects the vertical component of the projectile's motion, causing it to slow down as it ascends and speed up as it descends.
๐Ÿ’กHorizontal Acceleration
Horizontal acceleration is mentioned as being zero in the video, indicating no acceleration in the horizontal direction. This is typical for ideal projectile motion, where only gravity affects the vertical motion.
๐Ÿ’กCliff Height
The cliff height is the initial vertical position of the projectile, given as 95 meters in the video. It is an important parameter for calculating the total time of flight and the maximum height the projectile reaches above the ground.
๐Ÿ’กApex
The apex refers to the highest point in the trajectory of a projectile. The video discusses calculating the maximum height reached by the projectile at its apex, which is a key point for understanding the entire motion.
๐Ÿ’กTrajectory
Trajectory is the path that a projectile follows through the air. The video describes the projectile's trajectory as it is launched, reaches its apex, and then falls back down to the ground.
๐Ÿ’กTime of Flight
Time of flight is the total time a projectile spends in the air from launch to landing. The video calculates the time of flight as part of determining the overall motion of the projectile.
๐Ÿ’กHorizontal Distance
Horizontal distance is the total distance a projectile travels in the horizontal direction during its flight. The video calculates this to determine how far the projectile lands from the base of the cliff.
๐Ÿ’กResultant Velocity
Resultant velocity is the overall velocity of an object, considering both its horizontal and vertical components. The video calculates the resultant velocity just before the projectile hits the ground to understand its speed and direction of impact.
Highlights

A projectile is launched from a 95-meter high cliff with an initial speed of 15 meters per second at a 37-degree angle.

The initial velocity is given as 50 meters per second, which is a typographical error in the transcript.

The vertical acceleration due to gravity is -9.8 m/sยฒ, and the horizontal acceleration is 0 m/sยฒ.

The projectile's maximum height is calculated using the formula for vertical displacement in projectile motion.

The highest point reached by the projectile is 141 meters above the ground.

The total time in the air is determined by solving a quadratic equation for time.

The time to reach the bottom of the cliff is calculated to be 8.4 seconds.

The horizontal distance covered by the projectile is found using the horizontal velocity and time in the air.

The projectile covers a horizontal distance of 335 meters before hitting the ground.

The velocity components just before impact are calculated separately for the horizontal and vertical directions.

The vertical component of the velocity at impact is found to be -52 m/s, indicating a downward direction.

The horizontal component of the velocity remains constant at 40 m/s due to no horizontal acceleration.

The resultant velocity at the bottom of the cliff is calculated using the Pythagorean theorem.

The projectile's velocity just before impact is 66 m/s.

The problem-solving approach demonstrates the application of projectile motion equations in real-world scenarios.

The importance of accurate initial conditions for the correct calculation of projectile motion is highlighted.

The method illustrates the step-by-step process of solving for various aspects of projectile motion.

The transcript provides a clear example of how to apply physics formulas to calculate projectile motion parameters.

Transcripts
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