Study Guide: Motion in 2D, Projectiles, Inclined Planes, and Circular Motion — Step-by-Step Guidance

Section A: Vectors & Components (Lesson 6)

Q1. Why is velocity a vector but speed is not?

Background

Topic: Scalars vs. Vectors

This question tests your understanding of the difference between scalar and vector quantities in physics.

Key Terms:

• Vector: A quantity with both magnitude and direction (e.g., velocity, force).

• Scalar: A quantity with only magnitude (e.g., speed, mass).

Step-by-Step Guidance

1. Recall the definitions of speed and velocity. Think about what information each provides about an object's motion.

2. Consider whether direction is important for each quantity. Does speed tell you which way something is moving?

3. Think about how velocity is represented (e.g., as an arrow or with components), and how speed is represented (just a number).

Try solving on your own before revealing the answer!

Q2. If a force has a negative y-component, what does that physically mean?

Background

Topic: Vector Components

This question checks your understanding of vector direction and how components relate to physical direction in space.

Key Terms:

• Component: The projection of a vector along an axis (e.g., x or y).

• Negative y-component: The vector points in the negative y-direction (downward if y is vertical).

Step-by-Step Guidance

1. Recall the meaning of positive and negative directions on the coordinate axes.

2. Think about what it means for a vector's y-component to be negative. Which way is the force acting?

3. Visualize or sketch a vector with a negative y-component to see its direction.

Try solving on your own before revealing the answer!

Q3. A vector has magnitude 10 N and angle 30° above the +x axis. Is the x-component larger or smaller than the y-component? Why?

Background

Topic: Vector Components and Trigonometry

This question tests your ability to decompose a vector into x and y components using trigonometric functions.

Key Formulas:

• $F_x = F \cos \theta$

• $F_y = F \sin \theta$

Step-by-Step Guidance

1. Identify the given values: $F = 10\ \text{N}$, $\theta = 30^\circ$ above the +x axis.

2. Recall which trigonometric function gives the x-component and which gives the y-component.

3. Compare the values of $\cos 30^\circ$ and $\sin 30^\circ$ to determine which component is larger.

Try solving on your own before revealing the answer!

Q4. If two perpendicular force components are 6 N and 8 N, why is the resultant not 14 N?

Background

Topic: Vector Addition (Pythagorean Theorem)

This question tests your understanding of how to combine perpendicular vectors to find the resultant.

Key Formula:

• $F_{\text{net}} = \sqrt{F_1^2 + F_2^2}$

Step-by-Step Guidance

1. Recall that perpendicular vectors combine using the Pythagorean theorem, not simple addition.

2. Think about the geometric meaning: the resultant is the hypotenuse of a right triangle formed by the two components.

3. Set up the formula for the resultant using the given values (but do not calculate the final value).

Try solving on your own before revealing the answer!

Q5. Why do we use sine for the opposite side and cosine for the adjacent side in component decomposition?

Background

Topic: Trigonometry in Physics

This question checks your understanding of basic trigonometric relationships in right triangles as applied to vectors.

Key Terms:

• Opposite side: The side opposite the angle in question.

• Adjacent side: The side next to the angle in question.

• Hypotenuse: The longest side of a right triangle.

Step-by-Step Guidance

1. Recall the definitions of sine and cosine in a right triangle: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.

2. Think about how a vector's components form a right triangle with the vector as the hypotenuse.

3. Relate the sides of the triangle to the x and y components of the vector.

Try solving on your own before revealing the answer!

Q6. If a vector points into Quadrant III, what signs must its components have?

Background

Topic: Coordinate System and Vector Direction

This question tests your understanding of the signs of vector components in different quadrants.

Key Terms:

• Quadrant III: The region where both x and y are negative.

Step-by-Step Guidance

1. Recall the layout of the four quadrants in the Cartesian plane.

2. Determine the sign of the x-component and y-component for a vector in Quadrant III.

3. Think about what it means physically for both components to be negative.

Try solving on your own before revealing the answer!

Q7. If Fx = 0, can the object still be moving? Explain physically.

Background

Topic: Newton's Laws of Motion

This question checks your understanding of the relationship between force and motion, specifically Newton's First Law.

Key Terms:

• Net force ($F_x$): The sum of all forces in the x-direction.

• Inertia: The tendency of an object to maintain its state of motion.

Step-by-Step Guidance

1. Recall Newton's First Law: An object in motion stays in motion unless acted on by a net force.

2. Consider what happens if the net force in the x-direction is zero. Does this mean the object must be at rest?

3. Think about examples where an object moves at constant velocity with zero net force.

Try solving on your own before revealing the answer!