Q1. A projectile is launched with an initial velocity at an angle above the horizontal. Describe the motion and key equations for its trajectory.

Background

Topic: Kinematics of projectile motion

This question tests your understanding of how to analyze the motion of a projectile under constant acceleration due to gravity, including decomposing the initial velocity and using kinematic equations.

Key Terms and Formulas

• Projectile motion: The motion of an object thrown or projected into the air, subject only to gravity.

• Initial velocity components: ,

• Acceleration: , (where is the acceleration due to gravity)

• Kinematic equations for position: ,

Step-by-Step Guidance

1. Decompose the initial velocity into horizontal and vertical components using trigonometry: , .

2. Write the equations for horizontal and vertical positions as functions of time: , .

3. Identify the acceleration in each direction: (no horizontal acceleration), (downward acceleration due to gravity).

4. Set up the equations to solve for quantities like maximum height, range, or time of flight, depending on what is asked.

Try solving on your own before revealing the answer!

Q2. CNS-11: Consider a person standing in an elevator that is moving upward at constant speed. The magnitude of the upward normal force, , exerted by the elevator floor on the person’s feet is (ignore friction and similar fine details):

Background

Topic: Forces and Newton's Laws in vertical motion

This question tests your understanding of normal force and weight in situations involving elevators, and how Newton's laws apply when acceleration is zero or nonzero.

Key Terms and Formulas

• Normal force (): The force exerted by a surface perpendicular to the object.

• Weight (): , where is mass and is acceleration due to gravity.

• Newton's Second Law:

Step-by-Step Guidance

1. Identify the forces acting on the person: the upward normal force and the downward weight .

2. Write Newton's Second Law for vertical motion: .

3. For constant speed, , so .

4. Now consider the case where the elevator is accelerating upward: , so .

Try solving on your own before revealing the answer!

Q3. CNS-15: In a tilted coordinate system, the acceleration vector is always in the x-direction. The coordinate axes are at an angle to the horizontal. What is the x-component of the acceleration?

Background

Topic: Decomposition of vectors in tilted coordinate systems

This question tests your ability to resolve acceleration vectors into components along tilted axes, which is common in problems involving inclined planes.

Key Terms and Formulas

• Acceleration vector:

• Component along tilted axis: or depending on axis orientation

Step-by-Step Guidance

1. Draw the coordinate axes tilted at angle to the horizontal.

2. Express the acceleration vector in terms of its components along the tilted axes.

3. Use trigonometric relationships to find the x-component: or .

4. Check the orientation of the axes to determine which trigonometric function applies.

Try solving on your own before revealing the answer!

Q4. CNS-17: Atwood's machine is a pulley with two masses connected by a string as shown. The mass of A, , is twice the mass of B, . The tension in the string on the left, above mass A, is...

Background

Topic: Dynamics of systems with pulleys (Atwood's machine)

This question tests your understanding of tension forces and Newton's laws in systems with multiple masses and pulleys.

Key Terms and Formulas

• Tension (): The force transmitted through a string, rope, or cable.

• Newton's Second Law:

• Atwood's machine: A system with two masses connected by a string over a pulley.

Step-by-Step Guidance

1. Draw free-body diagrams for both masses, and .

2. Write Newton's Second Law for each mass: , .

3. Set up the equations and relate the accelerations (since the string is massless and inextensible).

4. Substitute and solve for in terms of , , and .

Try solving on your own before revealing the answer!

Q5. CNS-18: A mass is pulled along a rough table at constant velocity with an external force at some angle above the horizontal. The magnitudes of the forces on the free-body diagram have not been drawn carefully, but the directions of the forces are correct. Which statement below must be true?

Background

Topic: Forces and equilibrium with friction

This question tests your understanding of force balance and friction when an object moves at constant velocity.

Key Terms and Formulas

• Constant velocity: Net force is zero ()

• Friction force (): Opposes motion,

• Normal force (): Perpendicular to the surface

Step-by-Step Guidance

1. Draw the free-body diagram showing all forces: applied force , weight , normal force , and friction .

2. Write Newton's Second Law for both x and y directions. For constant velocity, .

3. Set up the equations: , .

4. Relate the friction force to the normal force: .

Try solving on your own before revealing the answer!