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Questions and Answers
An object with momentum $p$ and mass $m$ has kinetic energy expressed as $E_k = \frac{p^2}{2m}$. If the momentum $p$ is doubled while the mass $m$ is halved, by what factor does the kinetic energy $E_k$ change?
An object with momentum $p$ and mass $m$ has kinetic energy expressed as $E_k = \frac{p^2}{2m}$. If the momentum $p$ is doubled while the mass $m$ is halved, by what factor does the kinetic energy $E_k$ change?
- The kinetic energy increases by a factor of 2.
- The kinetic energy remains the same.
- The kinetic energy is reduced to $\frac{1}{2}$ of its original value.
- The kinetic energy increases by a factor of 8. (correct)
A cyclist is riding a bicycle with wheels of radius $r = 0.5$ meters. If the angular velocity of the wheels is $\omega = 6$ radians per second, what is the linear speed $v$ of the bicycle in meters per second?
A cyclist is riding a bicycle with wheels of radius $r = 0.5$ meters. If the angular velocity of the wheels is $\omega = 6$ radians per second, what is the linear speed $v$ of the bicycle in meters per second?
- $v = 1.5 \text{ m/s}$
- $v = 3 \text{ m/s}$ (correct)
- $v = 12 \text{ m/s}$
- $v = 6.5 \text{ m/s}$
An object moves in a circular path with a constant angular velocity. If its period is doubled, how does its angular velocity change?
An object moves in a circular path with a constant angular velocity. If its period is doubled, how does its angular velocity change?
- The angular velocity remains the same.
- The angular velocity quadruples.
- The angular velocity doubles.
- The angular velocity is halved. (correct)
A point on a rotating disk with a radius of 0.2 meters has a linear velocity of 4 m/s. What is the angular velocity of the disk in radians per second?
A point on a rotating disk with a radius of 0.2 meters has a linear velocity of 4 m/s. What is the angular velocity of the disk in radians per second?
What is the angular displacement in radians of an object that has completed 3 full revolutions around a circle?
What is the angular displacement in radians of an object that has completed 3 full revolutions around a circle?
An object is moving in a circular path. If the radius of the circle is doubled and the linear speed of the object remains the same, how does the angular velocity change?
An object is moving in a circular path. If the radius of the circle is doubled and the linear speed of the object remains the same, how does the angular velocity change?
Convert an angle of 270 degrees to radians.
Convert an angle of 270 degrees to radians.
Angular velocity is best described as...
Angular velocity is best described as...
In a collision where object A hits object B, and after the collision, object A has a negative velocity while object B has a positive velocity, what can be inferred about the directions of their movements?
In a collision where object A hits object B, and after the collision, object A has a negative velocity while object B has a positive velocity, what can be inferred about the directions of their movements?
A system consists of two colliding objects. If the total kinetic energy of the system decreases after the collision, but momentum is conserved, what type of collision occurred?
A system consists of two colliding objects. If the total kinetic energy of the system decreases after the collision, but momentum is conserved, what type of collision occurred?
In an isolated system, two objects collide and stick together. Which statement accurately describes this scenario?
In an isolated system, two objects collide and stick together. Which statement accurately describes this scenario?
In an explosion, the kinetic energy after the event is greater than before. Which of the following statements is most accurate regarding momentum conservation?
In an explosion, the kinetic energy after the event is greater than before. Which of the following statements is most accurate regarding momentum conservation?
A non-relativistic particle has a momentum of $p$ and a mass of $m$. How does its kinetic energy ($E_k$) change if its momentum is doubled while its mass remains constant?
A non-relativistic particle has a momentum of $p$ and a mass of $m$. How does its kinetic energy ($E_k$) change if its momentum is doubled while its mass remains constant?
A particle's kinetic energy is found to be $E_k$ using the formula $E_k = \frac{1}{2}mv^2$. If you determine the particle's velocity by rearranging the momentum formula $p = mv$, which expression correctly relates kinetic energy to momentum?
A particle's kinetic energy is found to be $E_k$ using the formula $E_k = \frac{1}{2}mv^2$. If you determine the particle's velocity by rearranging the momentum formula $p = mv$, which expression correctly relates kinetic energy to momentum?
Two balls collide. Ball A has mass 5kg and Ball B has mass 10kg. After the collision, the velocity of A is -5 m/s. Using equation [6]: 0 = 15v ′1 + 10√3 × v ′2 and the value of v’1, what is the value of v’2?
Two balls collide. Ball A has mass 5kg and Ball B has mass 10kg. After the collision, the velocity of A is -5 m/s. Using equation [6]: 0 = 15v ′1 + 10√3 × v ′2 and the value of v’1, what is the value of v’2?
Two balls collide. Ball A has mass 5kg and Ball B has mass 10kg. Using equation [5]: 150 = 15v ′1 + 30√3 × v ′2 and the value of v’2 as $\frac{5 \sqrt{3}}{2}$, what is the value of v’1?
Two balls collide. Ball A has mass 5kg and Ball B has mass 10kg. Using equation [5]: 150 = 15v ′1 + 30√3 × v ′2 and the value of v’2 as $\frac{5 \sqrt{3}}{2}$, what is the value of v’1?
An object moves in a circle of radius $r$ at a constant speed $v$. If the radius is doubled and the speed is halved, how does the centripetal acceleration change?
An object moves in a circle of radius $r$ at a constant speed $v$. If the radius is doubled and the speed is halved, how does the centripetal acceleration change?
Two similar triangles are constructed: one from displacements on a circular path and another from changes in velocity. If the ratio of corresponding sides in the displacement triangle is 1:2, what is the equivalent ratio in the velocity triangle, assuming constant speed?
Two similar triangles are constructed: one from displacements on a circular path and another from changes in velocity. If the ratio of corresponding sides in the displacement triangle is 1:2, what is the equivalent ratio in the velocity triangle, assuming constant speed?
An object is moving in a circular path with a constant angular velocity. What can be definitively stated about the object's linear velocity?
An object is moving in a circular path with a constant angular velocity. What can be definitively stated about the object's linear velocity?
Consider an object undergoing uniform circular motion. If the angular velocity is tripled and the radius is halved, by what factor does the centripetal acceleration change?
Consider an object undergoing uniform circular motion. If the angular velocity is tripled and the radius is halved, by what factor does the centripetal acceleration change?
An object moves in a circular path. At what point in the circle is the change in velocity vector ($\Delta v$) most closely aligned with the direction of the centripetal acceleration?
An object moves in a circular path. At what point in the circle is the change in velocity vector ($\Delta v$) most closely aligned with the direction of the centripetal acceleration?
Suppose a car is moving around a circular track. If the track is icy, reducing the maximum frictional force, and the car attempts to maintain the same speed, what is the likely outcome?
Suppose a car is moving around a circular track. If the track is icy, reducing the maximum frictional force, and the car attempts to maintain the same speed, what is the likely outcome?
An object is in uniform circular motion. If the period of the motion is doubled while the radius remains constant, what happens to the object's centripetal acceleration?
An object is in uniform circular motion. If the period of the motion is doubled while the radius remains constant, what happens to the object's centripetal acceleration?
Consider two particles moving in circular paths of the same radius. Particle A has twice the angular velocity of Particle B. What is the ratio of the centripetal force acting on Particle A to that acting on Particle B, assuming they have the same mass?
Consider two particles moving in circular paths of the same radius. Particle A has twice the angular velocity of Particle B. What is the ratio of the centripetal force acting on Particle A to that acting on Particle B, assuming they have the same mass?
Consider a collision where momentum is conserved. If the initial momentum of a two-body system is entirely in the x-direction, which statement regarding the final momentum components is correct?
Consider a collision where momentum is conserved. If the initial momentum of a two-body system is entirely in the x-direction, which statement regarding the final momentum components is correct?
In an elastic collision between two objects where one object is initially at rest, what is the maximum angle that the initially moving object can be scattered?
In an elastic collision between two objects where one object is initially at rest, what is the maximum angle that the initially moving object can be scattered?
Two balls with different masses collide. Ball 1 has a mass of $m_1$ and ball 2 has a mass of $m_2$. Initially, ball 1 moves with velocity $v_1$ and ball 2 is at rest. After the collision, both balls move at angles $\theta_1$ and $\theta_2$ with respect to the initial direction of ball 1. Which set of equations correctly describes the conservation of momentum in the x and y directions?
Two balls with different masses collide. Ball 1 has a mass of $m_1$ and ball 2 has a mass of $m_2$. Initially, ball 1 moves with velocity $v_1$ and ball 2 is at rest. After the collision, both balls move at angles $\theta_1$ and $\theta_2$ with respect to the initial direction of ball 1. Which set of equations correctly describes the conservation of momentum in the x and y directions?
A billiard ball collides with another identical ball initially at rest. After the collision, what can be said about the angle between their trajectories, assuming the collision is perfectly elastic and occurs on a frictionless surface?
A billiard ball collides with another identical ball initially at rest. After the collision, what can be said about the angle between their trajectories, assuming the collision is perfectly elastic and occurs on a frictionless surface?
Consider two objects colliding elastically. Object A has mass $m$ and object B has mass $3m$. Initially, object A moves at a speed of $v$ and object B is at rest. After the collision, both objects move away at some angle with respect to the initial direction of object A. What is the relationship between the speeds of the two objects after the collision?
Consider two objects colliding elastically. Object A has mass $m$ and object B has mass $3m$. Initially, object A moves at a speed of $v$ and object B is at rest. After the collision, both objects move away at some angle with respect to the initial direction of object A. What is the relationship between the speeds of the two objects after the collision?
A ball of mass $m_1$ is traveling at a velocity $v_1$ when it collides with a stationary ball of mass $m_2$. After the collision, $m_1$ comes to a complete stop. Assuming the collision is perfectly elastic, what is the velocity of $m_2$ after the collision?
A ball of mass $m_1$ is traveling at a velocity $v_1$ when it collides with a stationary ball of mass $m_2$. After the collision, $m_1$ comes to a complete stop. Assuming the collision is perfectly elastic, what is the velocity of $m_2$ after the collision?
Two balls of equal mass undergo an elastic collision. Ball A is initially moving with a velocity $v$, and ball B is at rest. If after the collision ball A moves off at an angle of 45 degrees with respect to its original direction, what is the angle of ball B with respect to the original direction of ball A?
Two balls of equal mass undergo an elastic collision. Ball A is initially moving with a velocity $v$, and ball B is at rest. If after the collision ball A moves off at an angle of 45 degrees with respect to its original direction, what is the angle of ball B with respect to the original direction of ball A?
In a perfectly inelastic collision between two objects, what quantity is necessarily conserved?
In a perfectly inelastic collision between two objects, what quantity is necessarily conserved?
A cricket ball with a mass of 0.16 kg is struck by a bat. Before the impact, the ball is traveling at 30 m/s. After the impact, the ball travels in the opposite direction at 40 m/s. If the bat exerts a constant force on the ball during impact and the impact lasts for 0.001 seconds, calculate the magnitude of the force exerted by the bat on the ball.
A cricket ball with a mass of 0.16 kg is struck by a bat. Before the impact, the ball is traveling at 30 m/s. After the impact, the ball travels in the opposite direction at 40 m/s. If the bat exerts a constant force on the ball during impact and the impact lasts for 0.001 seconds, calculate the magnitude of the force exerted by the bat on the ball.
Two objects, A and B, collide on a frictionless surface. Object A has a mass of 2 kg and an initial velocity of $\langle 5, 0 \rangle$ m/s. Object B has a mass of 3 kg and an initial velocity of $\langle -3, 0 \rangle$ m/s. After the collision, object A has a velocity of $\langle -1, 0 \rangle$ m/s. What is the final velocity of object B?
Two objects, A and B, collide on a frictionless surface. Object A has a mass of 2 kg and an initial velocity of $\langle 5, 0 \rangle$ m/s. Object B has a mass of 3 kg and an initial velocity of $\langle -3, 0 \rangle$ m/s. After the collision, object A has a velocity of $\langle -1, 0 \rangle$ m/s. What is the final velocity of object B?
A trolley of mass 3 kg is moving with a velocity of 5 m/s to the right. A second trolley, with a mass of 2 kg, is moving with a velocity of 2 m/s to the left. If the two trolleys collide and stick together, what is the magnitude and direction of their combined velocity immediately after the collision?
A trolley of mass 3 kg is moving with a velocity of 5 m/s to the right. A second trolley, with a mass of 2 kg, is moving with a velocity of 2 m/s to the left. If the two trolleys collide and stick together, what is the magnitude and direction of their combined velocity immediately after the collision?
A ball of mass $m$ is dropped from height $h$ onto a stationary heavy plate fixed to the ground and rebounds to a height of $0.64h$. What is the impulse imparted to the ball by the plate?
A ball of mass $m$ is dropped from height $h$ onto a stationary heavy plate fixed to the ground and rebounds to a height of $0.64h$. What is the impulse imparted to the ball by the plate?
A firework of mass 1 kg explodes into two pieces. One piece of mass 0.4 kg moves off at a velocity of $\langle 10, 20 \rangle$ m/s. If the other piece moves off at a velocity of $\langle -5, -10 \rangle$ m/s, what is its mass?
A firework of mass 1 kg explodes into two pieces. One piece of mass 0.4 kg moves off at a velocity of $\langle 10, 20 \rangle$ m/s. If the other piece moves off at a velocity of $\langle -5, -10 \rangle$ m/s, what is its mass?
A car of mass $m$ is traveling at a velocity $v$ when the driver applies the brakes, exerting a constant force $F$ to bring the car to a stop. What is the time duration $\Delta t$ during which the brakes are applied until the car comes to a complete stop?
A car of mass $m$ is traveling at a velocity $v$ when the driver applies the brakes, exerting a constant force $F$ to bring the car to a stop. What is the time duration $\Delta t$ during which the brakes are applied until the car comes to a complete stop?
A 2 kg ball is thrown at an angle of 30° above the horizontal with an initial speed of 10 m/s. Neglecting air resistance, what is the change in momentum of the ball after 1 second?
A 2 kg ball is thrown at an angle of 30° above the horizontal with an initial speed of 10 m/s. Neglecting air resistance, what is the change in momentum of the ball after 1 second?
Two identical billiard balls collide. Ball A is initially moving at $\langle 4, 0 \rangle$ m/s and ball B is stationary. After the collision, ball A moves at $\langle 2, 2 \rangle$ m/s. What is the velocity of ball B after the collision?
Two identical billiard balls collide. Ball A is initially moving at $\langle 4, 0 \rangle$ m/s and ball B is stationary. After the collision, ball A moves at $\langle 2, 2 \rangle$ m/s. What is the velocity of ball B after the collision?
Flashcards
Impulse
Impulse
Impulse is the change in momentum of an object.
Impulse equation
Impulse equation
Impulse is equal to the product of a constant force and the impact time (FΔt).
Newton's 2nd Law (Impulse)
Newton's 2nd Law (Impulse)
Newton's Second Law can be expressed as F = Δ(mv) / Δt.
Conservation of Momentum
Conservation of Momentum
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Momentum in 2D
Momentum in 2D
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2D Momentum Problems
2D Momentum Problems
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Momentum Before & After
Momentum Before & After
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Solving 2D Problems
Solving 2D Problems
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Momentum Equation (x-direction)
Momentum Equation (x-direction)
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Object at Rest
Object at Rest
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Resolving Vectors
Resolving Vectors
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Momentum Equation with Angles (x)
Momentum Equation with Angles (x)
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Momentum Equation (y-direction)
Momentum Equation (y-direction)
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Motion Along X-axis
Motion Along X-axis
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Momentum Equation with Angles (y)
Momentum Equation with Angles (y)
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Elastic Collision
Elastic Collision
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Inelastic Collision
Inelastic Collision
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Perfectly Inelastic Collision
Perfectly Inelastic Collision
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Explosion
Explosion
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Non-Relativistic Particle
Non-Relativistic Particle
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Kinetic Energy (Ek)
Kinetic Energy (Ek)
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Kinetic Energy Formula (non-relativistic)
Kinetic Energy Formula (non-relativistic)
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Momentum Formula
Momentum Formula
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Kinetic Energy (using momentum)
Kinetic Energy (using momentum)
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Radian
Radian
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Degrees to Radians
Degrees to Radians
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Angular Displacement (θ)
Angular Displacement (θ)
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Angular Velocity (ω)
Angular Velocity (ω)
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Angular Velocity Formula (ω)
Angular Velocity Formula (ω)
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Angular Velocity and Period
Angular Velocity and Period
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Centripetal Acceleration
Centripetal Acceleration
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Acceleration
Acceleration
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Circular Motion
Circular Motion
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v = ωr
v = ωr
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Similar Triangles
Similar Triangles
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a = v²/r
a = v²/r
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a = rω²
a = rω²
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Constantly changing velocity
Constantly changing velocity
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Study Notes
Impulse
- Newton's second law is expressed as F = ma
- Therefore F = Δ(mv) / Δt, because a = Δv / Δt
- This can be rearranged to F∆t = ∆(mv) or F∆t = ∆p
F∆trepresents impulse- Impulse signifies the change in momentum
- For example, a ball hit with a baseball bat with a force of 100 N and an impact time of 0.5 s results in a change in momentum
- The change in momentum is calculated as: 100 x 0.5 = 50 kgm/s
Conservation of Linear Momentum in Two Dimensions
- Momentum is conserved in interactions without external forces
- Momentum before equals momentum after an event like a collision
- Momentum conservation can vary along different dimensions
- Solving conservation of momentum problems in two dimensions involves resolving motion into components along perpendicular axes
- Then solving the resultant pair of problems in one dimension simultaneously
- When billiard balls collide, the total momentum before equals total momentum after, represented as m1V1x + m2V2x = m₁v'1x + m2v'2x for the x-axis
- Where m₁/m2 is mass, V₁x/V2x is initial velocity, and v'1x/ v'2x is the final velocity in the x-direction
- If ball 2 is initially at rest (v2x = 0): m1V1x = m₁v'1x + m2V'2x
- If ball 1 moves in the x-direction, V1x = v1, modifying the equation to m1V1 = m₁v'1x + m2v'2x
- Final velocities (v'₁/ v’2) are resolved into x and y components using trigonometry
- This may be used alongside the equation derived by considering the y-direction, in order to solve for unknown values.
- Total momentum before = Total momentum after, represented as M₁V1y + M2V2y = m₁v'1y + m2V'2y
- Ball 2 starts at rest(V2y = 0). Ball 1 moves along the x-axis(V1y =0). The above equation then becomes: 0 = m₁v'1y + m2V'2y
- The final velocities can be used to solve for unknown values.
- In a scenario with m₁ = 5 kg, m₂ = 10 kg, v₁ = 5 ms⁻¹, v₂ = 0 ms⁻¹, θ₁ = 60°, and θ₂ = 30°, equations are derived to find final velocities v'₁ and v'₂
- By substituting known values, simultaneous equations can be formulated and then solved: 25 = 2.5v'₁ + 8.66v'₂ and 0 = 4.33v'₁ + 5v'₂ and used to find final velocities
Elastic and Inelastic Collisions
- Elastic collisions conserve both momentum and kinetic energy
- Inelastic collisions conserve only momentum
- Some kinetic energy converts to other forms, possibly larger or smaller post-collision
- If colliding objects stick, it's an inelastic collision
- An explosion exemplifies an inelastic collision where kinetic energy increases
Kinetic Energy of a Non-Relativistic Particle
- For non-relativistic particles (speeds below light), kinetic energy (Ek) uses the formula: Ek = p²/2m
- This formula derives from kinetic energy (Ek = 0.5mv²) and momentum (p = mv) formulas
- By rearranging the momentum formula to v = p/m and substituting: Ek = 0.5 x m x (p/m)² = p²/2m
Angular Displacement and Radians
- Radians measure angles, defined where arc length equals the circle's radius
- A full circle's arc length (2πr) divided by its radius (r) = 2π radians
- Degrees convert to radians by multiplying by π/180
- Radians convert to degrees by multiplying by 180/π
Angular Velocity
- Angular displacement (θ) measures the angle an object turns in a direction, in radians or degrees
- Angular velocity (ω) measures the angle an object moves per time: calculated as ω = v/r with linear velocity (v) and radius (r)
- This rearranges to v = ωr
- Angular velocity is the angle in a circle (2π) divided by time period (T) : ω = 2π/T
- This rearranges to T = 2π/ω
Centripetal Acceleration
- Centripetal acceleration occurs in circular motion
- The formula is derived using vector diagrams
- For an object moving at speed v in a circular path with radius r, the triangles formed by the circular path and velocity vectors are similar; each has two sides of equal length (r/v)
- As v₁ is perpendicular to AB, a right-angled triangle is formed
- After rearranging derived values: a = v²/r = rω² (with v = ωr)
Circular Motion and Centripetal Force
- Constant speed circular motion involves constantly changing velocity (direction), indicating acceleration (centripetal acceleration)
- Newton's first law states acceleration requires a resultant force, known as centripetal force, directed towards the circle's center
- Centripetal force is required to produce and maintain circular motion
Centripetal Force
- Applying Newton's second law (F = ma), the centripetal force formula: F = ma = mv²/r = mrω²
- Where m is object mass, v is linear speed, r is path radius, and ω is angular velocity
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Description
A quiz covering concepts of circular motion, rotational dynamics, kinetic energy, angular velocity, and angular displacement. This includes calculating linear speed, and understanding the relationships between momentum, mass, and kinetic energy.
