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Questions and Answers
A projectile is launched at an angle of 30 degrees above the horizontal with an initial velocity of 20 m/s. What is the maximum height reached by the projectile?
A projectile is launched at an angle of 30 degrees above the horizontal with an initial velocity of 20 m/s. What is the maximum height reached by the projectile?
A projectile is launched horizontally from a cliff with an initial velocity of 10 m/s. If the cliff is 20 meters high, how long does it take for the projectile to hit the ground?
A projectile is launched horizontally from a cliff with an initial velocity of 10 m/s. If the cliff is 20 meters high, how long does it take for the projectile to hit the ground?
Two projectiles are launched with the same initial speed but at different angles. Which projectile will have the longer range?
Two projectiles are launched with the same initial speed but at different angles. Which projectile will have the longer range?
A projectile is launched at an angle of 45 degrees above the horizontal with an initial velocity of 30 m/s. What is the horizontal velocity of the projectile at its highest point?
A projectile is launched at an angle of 45 degrees above the horizontal with an initial velocity of 30 m/s. What is the horizontal velocity of the projectile at its highest point?
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A projectile is launched at an angle of 60 degrees above the horizontal with an initial velocity of 15 m/s. What is the vertical velocity of the projectile 2 seconds after launch?
A projectile is launched at an angle of 60 degrees above the horizontal with an initial velocity of 15 m/s. What is the vertical velocity of the projectile 2 seconds after launch?
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A ball is thrown horizontally off a cliff with an initial speed of 10 m/s. What is the ball's horizontal velocity 2 seconds later?
A ball is thrown horizontally off a cliff with an initial speed of 10 m/s. What is the ball's horizontal velocity 2 seconds later?
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A projectile is launched at an angle of 30 degrees above the horizontal. Which of the following statements about the projectile's motion is true?
A projectile is launched at an angle of 30 degrees above the horizontal. Which of the following statements about the projectile's motion is true?
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Which of the following is NOT a factor that affects the range of a projectile?
Which of the following is NOT a factor that affects the range of a projectile?
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Flashcards
Position Vector
Position Vector
A vector from the origin to point P in 2D.
Displacement Vector
Displacement Vector
The difference between position vectors at two times: Δ𝐫⃗ =𝐫⃗ (𝑡2)−𝐫⃗ (𝑡1).
Instantaneous Velocity
Instantaneous Velocity
The limit of the displacement vector as time approaches zero: 𝐯⃗ (𝑡)=limΔ𝑡→0 (Δ𝐫⃗)/Δt.
Average Velocity
Average Velocity
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Maximum Height of Projectile
Maximum Height of Projectile
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Time of Flight
Time of Flight
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Trajectory Equation
Trajectory Equation
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Horizontal Motion
Horizontal Motion
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Study Notes
Position, Velocity, and Displacement Vectors
- Position vector 𝐫⃗ (𝑡) describes a particle's location at time 𝑡 in 3D space. It's expressed as 𝑥(𝑡)𝐢ˆ+𝑦(𝑡)𝐣ˆ+𝑧(𝑡)𝐤ˆ, where 𝐢ˆ, 𝐣ˆ, and 𝐤ˆ are unit vectors.
- Displacement vector Δ𝐫⃗ shows the change in position from time 𝑡₁ to 𝑡₂ and is calculated as Δ𝐫⃗ =𝐫⃗ (𝑡₂)−𝐫⃗ (𝑡₁).
- Instantaneous velocity vector 𝐯⃗ (𝑡) is the limit of the average velocity as the time interval approaches zero: 𝐯⃗ (𝑡)=limΔ𝑡→0 (𝐫⃗ (𝑡+Δ𝑡)−𝐫⃗ (𝑡))/Δt = 𝑑𝐫⃗ 𝑑𝑡.
- Average velocity 𝐯⃗ avg is calculated as 𝐯⃗ avg=𝐫⃗ (𝑡₂)−𝐫⃗ (𝑡₁)/𝑡₂−𝑡₁.
- Components of 𝐯⃗ (𝑡) are 𝑣𝑥(𝑡)=𝑑𝑥(𝑡)/𝑑𝑡, 𝑣𝑦(𝑡)=𝑑𝑦(𝑡)/𝑑𝑡, 𝑣𝑧(𝑡)=𝑑𝑧(𝑡)/𝑑𝑡.
Projectile Motion
Maximum Height
- Maximum height (ℎ) of a projectile depends only on the vertical component of initial velocity (𝑣₀𝑦): ℎ=((𝑣₀𝑦)^2)/(2𝑔).
Time of Flight
- Total flight time (𝑇ₜₒf) is given by: 𝑇ₜₒf=2(𝑣₀sin𝜃₀)/𝑔.
Trajectory Equation
- Trajectory (𝑦(𝑥)) is described by: 𝑦=(tan𝜃₀)𝑥−[𝑔/2(𝑣₀cos𝜃₀)^2]𝑥^2 (in the form 𝑦=𝑎𝑥+𝑏𝑥^2). Where a=(tan𝜃₀) and b =−[𝑔/2(𝑣₀cos𝜃₀)^2] .
Range
- Range (𝑅), or horizontal distance traveled, can be calculated as: 𝑅=𝑣₀^2sin2𝜃₀/𝑔.
Horizontal and Vertical Motion (Special Cases)
- Completely Horizontal Launch: If launched horizontally, initial vertical velocity 𝑣₀𝑦 is zero, and vertical motion is governed by gravity alone.
- Horizontal Velocity: Horizontal velocity (𝑣₀𝑥) remains constant throughout the flight.
- Vertical Velocity: Vertical velocity (𝑣𝑦) changes due to gravity. At maximum height (apex), 𝑣𝑦=0.
- Time to Ground: Time to hit the ground depends solely on vertical velocity (neglecting air resistance), usually found via vertical motion equations.
Equations of motion (Horizontal and Vertical Components)
- Horizontal motion: 𝑣₀𝑥=𝑣𝑥,𝑥=𝑥₀+𝑣𝑥𝑡
- Vertical motion: 𝑦=𝑦₀+1/2(𝑣₀𝑦+𝑣𝑦)𝑡, 𝑣𝑦=𝑣₀𝑦−𝑔𝑡, 𝑦=𝑦₀+𝑣₀𝑦𝑡−1/2𝑔𝑡^2, 𝑣^2𝑦=𝑣₀^2𝑦−2𝑔(𝑦−𝑦₀).
Key Concepts
- Acceleration: The only acceleration affecting projectile motion is gravity (𝑔), acting vertically downwards.
- Symmetry: Upward and downward portions of the parabolic trajectory exhibit mirrored characteristics (velocity, displacement). However, signs change.
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Description
This quiz covers the concepts of position, velocity, and displacement vectors in three-dimensional space, as well as key principles of projectile motion, including maximum height and time of flight. Test your understanding on how these vectors are calculated and applied in various physical scenarios.
