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Questions and Answers
What is the initial speed of the boat before it is subjected to acceleration?
What is the initial speed of the boat before it is subjected to acceleration?
From what state does the rocket begin its motion?
From what state does the rocket begin its motion?
What type of graph can be constructed from the rocket's motion over the time interval 0 to 14 seconds?
What type of graph can be constructed from the rocket's motion over the time interval 0 to 14 seconds?
What does the position vector r represent in curvilinear motion?
What does the position vector r represent in curvilinear motion?
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When analyzing erratic motion, at what point is the boat said to stop?
When analyzing erratic motion, at what point is the boat said to stop?
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What does the equation v = ds / dt represent?
What does the equation v = ds / dt represent?
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How can a v-t graph be constructed from an a-t graph?
How can a v-t graph be constructed from an a-t graph?
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What is required to determine the velocity at a specific point from a s-t graph?
What is required to determine the velocity at a specific point from a s-t graph?
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What does the equation a = dv / dt indicate?
What does the equation a = dv / dt indicate?
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In the context of motion, what does the term 'erratic' imply?
In the context of motion, what does the term 'erratic' imply?
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What does the area under an a-t graph represent?
What does the area under an a-t graph represent?
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Which equation relates acceleration to velocity and displacement?
Which equation relates acceleration to velocity and displacement?
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Which variable must be known to estimate other unknown variables in kinematics?
Which variable must be known to estimate other unknown variables in kinematics?
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What is the relationship between the radial velocity vr and the radius r given in the equations?
What is the relationship between the radial velocity vr and the radius r given in the equations?
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What does the equation $a_r = r - rθ^2$ represent?
What does the equation $a_r = r - rθ^2$ represent?
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If one particle moves downwards, what happens to the other particle according to the motion analysis mentioned?
If one particle moves downwards, what happens to the other particle according to the motion analysis mentioned?
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What represents the instantaneous velocity of a particle?
What represents the instantaneous velocity of a particle?
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What remains constant during the motion of the particles in the example provided?
What remains constant during the motion of the particles in the example provided?
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What is the mathematical expression for instantaneous acceleration?
What is the mathematical expression for instantaneous acceleration?
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The equation $a_θ = rθ + 2rθ$ simplifies to which of the following?
The equation $a_θ = rθ + 2rθ$ simplifies to which of the following?
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Which of the following correctly expresses speed?
Which of the following correctly expresses speed?
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What does the term 'aavg' represent in the context of motion?
What does the term 'aavg' represent in the context of motion?
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What causes the change in velocity (∆v) of a particle?
What causes the change in velocity (∆v) of a particle?
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Which statement best describes a hodograph?
Which statement best describes a hodograph?
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What is the relationship between ∆r and ∆s as ∆t approaches zero?
What is the relationship between ∆r and ∆s as ∆t approaches zero?
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What happens to the average velocity as time interval approaches zero?
What happens to the average velocity as time interval approaches zero?
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What does the equation $a = a_x + a_y + a_z$ represent in the context of motion?
What does the equation $a = a_x + a_y + a_z$ represent in the context of motion?
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At $t=2$ seconds, what is the horizontal position of the weather balloon if $x=2.4t$?
At $t=2$ seconds, what is the horizontal position of the weather balloon if $x=2.4t$?
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What is the significance of the components $(v_0)_x$ and $(v_0)_y$ in projectile motion?
What is the significance of the components $(v_0)_x$ and $(v_0)_y$ in projectile motion?
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What is indicated by the equation $y=x^{2/3}$ for the balloon's path?
What is indicated by the equation $y=x^{2/3}$ for the balloon's path?
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For projectile motion with zero horizontal acceleration, which equation is applied?
For projectile motion with zero horizontal acceleration, which equation is applied?
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In the given equations, what does $a_x = v_x = x$ imply about the variables?
In the given equations, what does $a_x = v_x = x$ imply about the variables?
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At $t=2$ seconds, if $y=x^{2/3}$, what is the value of y given the previously calculated x?
At $t=2$ seconds, if $y=x^{2/3}$, what is the value of y given the previously calculated x?
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What is the definition of a direction vector in the context of motion?
What is the definition of a direction vector in the context of motion?
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Study Notes
Rectilinear Erratic Motion
- The motion of an object cannot be described by a single continuous mathematical function, instead, a series of equations are needed
- velocity, acceleration, and displacement are described
- v = ds/dt, a = dv/dt, ads = vdv
- The s-t graph shows the relationship between displacement and time
- The slope of the s-t graph is the velocity
- The v-t graph is constructed by plotting the velocity at each instant of time
- The slope of the v-t graph is the acceleration
- The a-t graph can be constructed similarly to the v-t graph
- If the a-t graph is given, the v-t graph can be determined by calculating the area under the a-t graph
- ∆v = ∫ a dt
- The change in velocity is equal to the area under the a-t graph
Work Examples
- To determine the maximum speed and stopping time of a boat given its acceleration, use the equations of motion.
- For example, if the boat starts from rest, the initial velocity is zero
- To determine the v-t and s-t graphs for a rocket given its acceleration, use the equations of motion
- The rocket is starting from rest
Curvilinear Motion
- The position of a particle is defined by the position vector r = r(t)
- The velocity of a particle is the rate of change of position with respect to time:
- v = lim (∆r/∆t) = lim (∆s/∆t) as ∆t -> 0
- v = ds/dt
- The direction of the velocity vector is tangent to the path of motion
- The acceleration of a particle is the rate of change of velocity with respect to time:
- a = lim (∆v/∆t) as ∆t ->0
- a = dv/dt
- The acceleration vector has components in the x, y, and z directions:
- a = axi + ayj + azk
Curvilinear Motion: Rectangular 3-dimension
- The horizontal position of a weather balloon can be defined by an equation x = 2.4t (m), where t is in seconds.
- The magnitude and direction of the balloon's velocity and acceleration can be determined using the equations of motion.
- Use the equation of the path y=x2/3 to find the y-component of the velocity and acceleration.
- The balloon's velocity and acceleration at t = 2s can be found using these equations.
Curvilinear Motion: Projectile Motion
- A projectile launched at point (x0, y0) with an initial velocity v0 has components (v0)x and (v0)y.
- The horizontal motion of the projectile is constant acceleration motion because the horizontal acceleration is zero (ax = 0).
- The vertical motion of the projectile is affected by gravity.
- The vertical acceleration is -g, which is the acceleration due to gravity.
- The equations of motion for projectile motion can be used to determine the trajectory of the projectile.
Dependent Motion Analysis of Two Particles
- When two particles' motions are interdependent, the equations of motion can be used to analyze the relationship between their positions, velocities, and accelerations.
- For example, two particles connected by a chord, where the length of the chord remains constant.
- The motion of one particle affects the motion of the other.
- As one particle moves up, the other moves to the left, and their velocities are related. The time derivatives of the lengths of the chord segments can be used to determine the relationship.
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Description
Explore the principles of rectilinear erratic motion through this quiz. Understand how to analyze displacement, velocity, and acceleration using various graphs. Test your ability to apply equations of motion to real-world examples.
