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Questions and Answers
A projectile is launched with an initial velocity $v_o$ at an angle $\theta$ with respect to the horizontal. Assuming negligible air resistance, what is the primary factor that connects the horizontal and vertical components of its motion?
A projectile is launched with an initial velocity $v_o$ at an angle $\theta$ with respect to the horizontal. Assuming negligible air resistance, what is the primary factor that connects the horizontal and vertical components of its motion?
- The projectile's mass.
- The acceleration due to gravity.
- Time. (correct)
- The launch angle $\theta$.
A ball is thrown off a cliff with an initial horizontal velocity. Which of the following statements best describes how the horizontal and vertical components of its velocity change over time (assuming air resistance is negligible)?
A ball is thrown off a cliff with an initial horizontal velocity. Which of the following statements best describes how the horizontal and vertical components of its velocity change over time (assuming air resistance is negligible)?
- The horizontal component remains constant, while the vertical component increases at a constant rate. (correct)
- Both the horizontal and vertical components remain constant.
- Both the horizontal and vertical components increase at a constant rate.
- The horizontal component decreases at a constant rate, while the vertical component increases at a constant rate.
Two vectors, A and B, have magnitudes of 5 units and 8 units, respectively. Which of the following is NOT a possible magnitude for the resultant vector when A and B are added?
Two vectors, A and B, have magnitudes of 5 units and 8 units, respectively. Which of the following is NOT a possible magnitude for the resultant vector when A and B are added?
- 12 units
- 7 units
- 15 units (correct)
- 3 units
A car is traveling at a constant velocity. Suddenly, it enters a muddy patch where it experiences constant deceleration. Which of the following statements is true?
A car is traveling at a constant velocity. Suddenly, it enters a muddy patch where it experiences constant deceleration. Which of the following statements is true?
Vector A has components $A_x = 4$ and $A_y = -3$. What is the angle of this vector with respect to the positive x-axis?
Vector A has components $A_x = 4$ and $A_y = -3$. What is the angle of this vector with respect to the positive x-axis?
An object undergoes motion with constant acceleration. If its initial velocity components are $U_{xo} = 10 m/s$ and $U_{yo} = 5 m/s$, and its acceleration components are $a_x = -2 m/s^2$ and $a_y = 0 m/s^2$, what is the x-component of the object's velocity after 3 seconds?
An object undergoes motion with constant acceleration. If its initial velocity components are $U_{xo} = 10 m/s$ and $U_{yo} = 5 m/s$, and its acceleration components are $a_x = -2 m/s^2$ and $a_y = 0 m/s^2$, what is the x-component of the object's velocity after 3 seconds?
You are piloting a small aircraft, and the GPS indicates that your velocity relative to the ground is 185 km/h at an angle of 33.0 to the right of the forward direction, while your airplane's velocity relative to the air is 242 km/h straight ahead. What is the wind velocity relative to the ground?
You are piloting a small aircraft, and the GPS indicates that your velocity relative to the ground is 185 km/h at an angle of 33.0 to the right of the forward direction, while your airplane's velocity relative to the air is 242 km/h straight ahead. What is the wind velocity relative to the ground?
A motor boat is crossing a river that is flowing east at a speed of 5 m/s. The boat maintains a constant heading due north at a speed of 10 m/s relative to the water. What is the magnitude of the boat's resultant velocity with respect to an observer standing on the riverbank?
A motor boat is crossing a river that is flowing east at a speed of 5 m/s. The boat maintains a constant heading due north at a speed of 10 m/s relative to the water. What is the magnitude of the boat's resultant velocity with respect to an observer standing on the riverbank?
Flashcards
Components of Motion
Components of Motion
Motion in two dimensions analyzed through linear components tied by time.
Initial Velocity Components
Initial Velocity Components
Components of initial velocity are given by $U_{xo} = v_o cos θ$ and $U_{yo} = v_o sin θ$.
Displacement Equation
Displacement Equation
Displacement with constant acceleration: $x = x_o + U_{xo}t + rac{1}{2}a_xt^2$ and $y= y_o + U_{yo}t + rac{1}{2}a_yt^2$.
Magnitude of Velocity
Magnitude of Velocity
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Vector Addition Method
Vector Addition Method
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Vector Subtraction
Vector Subtraction
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Equality of Vectors
Equality of Vectors
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Components of a Vector
Components of a Vector
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Study Notes
Motion in Two Dimensions
- Motion in two dimensions is analyzed by considering the components of linear motion. Time connects these components.
- Components of Initial Velocity:
- Ux0 = vo cos θ
- Uyo = vo sin θ
- Components of Displacement (constant acceleration only):
- x = xo + Ux0t + ½axt²
- y = yo + Vy0t + ½ayt²
- Straight-line distance (origin to (x,y)): d = √x² + y²
- Magnitude of displacement is d
- Components of Velocity (constant acceleration only):
- Ux = Ux0 + axt
- Uy = Vy0 + ayt
- Magnitude of velocity: v = √vₓ² + vᵧ²
- Direction of velocity: θ=tan⁻¹(vᵧ/vₓ) relative to the x-axis
- If relative to the x-axis:
- 0 > 45° Uyo > Uxo
- 0 = 45° Uyo = Uxo
- 0 < 45° Uyo < Uxo
Remarks
- Constant velocity (a = 0): Motion in a straight line.
- Acceleration in direction of velocity or opposite to it: Motion in a straight line.
- Acceleration at an angle other than 0° or 180° to velocity: Motion in a curved path.
Vector Addition and Subtraction
- Vector addition (Geometric Methods):
- Triangle Method: Tail-to-Tip method.
- Resultant vector (R): Goes from the tail of the first vector to the tip of the last vector.
- Sum for more than two vectors: Polygon method.
- Vector subtraction: A – B = A + (-B)
- Important Note: The same units are necessary when adding vectors.
- Vector Addition is Commutative: A + B = B + A
- Vector Addition is Associative: (A + B) + C = A + (B + C)
- Equality of Two Vectors: Vectors A and B are equal if they have the same magnitude and the same direction.
- Vector Representation:
- Components (rectangular components): Cx = C cos θ; Cy = C sin θ
- Magnitude-angle form: C = √Cₓ² + Cᵧ²; θ = tan⁻¹(Cᵧ/Cₓ)
- Example: Magnitude of C is calculated using the Pythagorean theorem. Orientation of C relative to the x-axis by angle θ.
Procedures for Adding Vectors by Component Method
- Resolve vectors into x- and y-components.
- Add x-components together, and y-components together.
- Express the resultant using either:
- Component form (e.g., C = Cxı + Cyj)
- Magnitude-angle form (C = √Cₓ² + Cᵧ² , θ = tan⁻¹(Cᵧ/Cₓ))
Other Topics (from the text)
- Finding the vector given displacement and velocity information.
- Follow-up exercises and examples are present in the text, for further practice and clarity.
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Description
Explore motion in two dimensions by analyzing linear motion components connected by time. Understand initial velocity, displacement, and velocity with constant acceleration. Learn to calculate straight-line distance and velocity direction.
