Q1. The density of silver is 10.5 g/cm3. What is this density in SI units?

Background

Topic: Unit Conversion, Density

This question tests your ability to convert density from cgs units (grams per cubic centimeter) to SI units (kilograms per cubic meter).

Key Terms and Formulas:

• Density (): , where is mass and is volume.

• 1 g = kg

• 1 cm = m$^3$

Step-by-Step Guidance

1. Start by writing the given density: .

2. Convert grams to kilograms: .

3. Convert cubic centimeters to cubic meters: .

4. Set up the conversion so that units cancel appropriately, and express the density in .

Try solving on your own before revealing the answer!

Q2. The volume of a solid cylinder is given by . If the radius is doubled and the height is halved, by what factor does the mass change? (Assume constant density.)

Background

Topic: Scaling Laws, Volume and Mass

This question tests your understanding of how changes in dimensions affect volume and mass, given constant density.

Key Terms and Formulas:

• Volume of a cylinder:

• Mass: (where is density)

Step-by-Step Guidance

1. Write the original volume: .

2. Express the new radius and height: , .

3. Substitute the new values into the volume formula: .

4. Simplify the expression to find in terms of .

5. Since density is constant, mass changes by the same factor as volume.

Try solving on your own before revealing the answer!

Q3. Vector is 3.0 cm at above the horizontal. Vector is 6.0 cm at below the horizontal. Use and . What is ?

Background

Topic: Vector Addition, Components

This question tests your ability to break vectors into components, add them, and find the magnitude of the resultant vector.

Key Terms and Formulas:

• Vector components: ,

• Magnitude:

Step-by-Step Guidance

1. Find 's components: , .

2. Find 's components: , (negative because it's below horizontal).

3. Add the and components: , .

4. Set up the magnitude formula: .

Try solving on your own before revealing the answer!

Q4. For the vectors in the previous question, what angle does make with the axis?

Background

Topic: Vector Operations, Direction Angles

This question tests your ability to perform vector operations and find the direction of a resultant vector using inverse tangent.

Key Terms and Formulas:

• Resultant components: ,

• Angle:

Step-by-Step Guidance

1. Calculate and using the components from Q3.

2. Subtract and from and respectively to get and .

3. Set up the formula for the angle: .

4. Plug in the expressions for and using the given trigonometric values.

Try solving on your own before revealing the answer!

Q5. Vector has magnitude 5.0 m and makes an angle of with the positive x-axis. Write $\vec{A}$ in terms of and .

Background

Topic: Vector Components

This question tests your ability to express a vector in component form using unit vectors.

Key Terms and Formulas:

Step-by-Step Guidance

1. Calculate .

2. Calculate .

3. Write in terms of and using these components.

Try solving on your own before revealing the answer!

Q6. A rocket starts from rest. For the first 10.0 s its acceleration is (m/s). How high is the rocket at s?

Background

Topic: Kinematics with Variable Acceleration

This question tests your ability to integrate acceleration to find velocity and position when acceleration is a function of time.

Key Terms and Formulas:

• Acceleration:

• Velocity:

• Position:

Step-by-Step Guidance

1. Integrate with respect to to find , using the initial condition that the rocket starts from rest.

2. Integrate to find , using the initial condition that the rocket starts from .

3. Set up the expression for using your result from the previous step.

Try solving on your own before revealing the answer!

Q7. The drive between San Diego and Los Angeles takes 2 h at 100 km/h. If heavy traffic reduces the average speed to 72 km/h, how much longer does the trip take?

Background

Topic: Average Speed, Time Calculation

This question tests your ability to relate distance, speed, and time, and to compare travel times at different speeds.

Key Terms and Formulas:

• Distance:

• Time:

Step-by-Step Guidance

1. Calculate the distance between the cities using the original speed and time.

2. Calculate the new travel time using the reduced speed.

3. Find the difference between the new and original travel times.

Try solving on your own before revealing the answer!

Q8. A bird’s position is . Find the instantaneous velocity at s.

Background

Topic: Instantaneous Velocity, Differentiation

This question tests your ability to find the instantaneous velocity by differentiating the position function with respect to time.

Key Terms and Formulas:

• Instantaneous velocity:

Step-by-Step Guidance

1. Differentiate with respect to to get .

2. Substitute s into your expression for .

Try solving on your own before revealing the answer!

Q9. A car accelerates from rest to 72 km/h in 10 s. Compute the average acceleration during this interval.

Background

Topic: Average Acceleration, Kinematics

This question tests your ability to calculate average acceleration from a change in velocity over a time interval.

Key Terms and Formulas:

• Average acceleration:

• Convert km/h to m/s:

Step-by-Step Guidance

1. Convert the final velocity from km/h to m/s.

2. Use the formula for average acceleration with the initial velocity as zero.

3. Set up the calculation for using the converted velocity and time interval.

Try solving on your own before revealing the answer!

Q10. A shot is thrown at 12.0 m/s at above horizontal. It lands 2.00 s later. How far does it travel horizontally?

Background

Topic: Projectile Motion

This question tests your ability to analyze projectile motion and calculate horizontal displacement.

Key Terms and Formulas:

• Horizontal velocity:

• Horizontal distance:

Step-by-Step Guidance

1. Calculate the horizontal component of the initial velocity using .

2. Multiply the horizontal velocity by the total time of flight to get the horizontal distance.

Try solving on your own before revealing the answer!

Q11. A 1.0 m string makes a constant angle of with the vertical while a ball moves in a horizontal circle. The period is 2.0 s. Find the radial acceleration. Leave your answer in terms of .

Background

Topic: Circular Motion, Centripetal Acceleration

This question tests your ability to calculate centripetal (radial) acceleration for an object in uniform circular motion.

Key Terms and Formulas:

• Radius: (where is the string length)

• Speed:

• Radial acceleration:

Step-by-Step Guidance

1. Calculate the radius of the circle using the string length and angle.

2. Find the speed of the ball using the period and radius.

3. Set up the formula for radial acceleration in terms of .

Try solving on your own before revealing the answer!

Q12. A moving sidewalk travels at 1.5 m/s and is 30.0 m long. A woman walks at 2.0 m/s relative to the sidewalk opposite its motion. How long does she take to cross?

Background

Topic: Relative Velocity

This question tests your understanding of relative velocities and how to calculate time to cover a distance when two velocities are in opposite directions.

Key Terms and Formulas:

• Relative velocity: (since she walks opposite to the sidewalk)

• Time:

Step-by-Step Guidance

1. Determine the woman's velocity relative to the ground by subtracting her walking speed from the sidewalk speed.

2. Set up the formula for time using the total distance and relative velocity.

Try solving on your own before revealing the answer!

Free Response 1(a). Two stones are thrown vertically upward from the ground, one with three times the initial speed of the other. If the faster stone takes 10 s to return to the ground, how long does the slower stone take to return?

Background

Topic: 1D Kinematics, Free Fall

This question tests your understanding of projectile motion under gravity and how time of flight depends on initial velocity.

Key Terms and Formulas:

• Time of flight for vertical motion:

• Relationship between initial velocities:

Step-by-Step Guidance

1. Write the time of flight for each stone in terms of its initial velocity.

2. Set up the ratio of times using the given relationship between initial velocities.

3. Use the given time for the faster stone to solve for the slower stone's time.

Try solving on your own before revealing the answer!

Free Response 1(b). If the slower stone reaches a maximum height , how high (in terms of $H$) does the faster stone rise?

Background

Topic: Kinematics, Maximum Height

This question tests your ability to relate maximum heights to initial velocities in projectile motion.

Key Terms and Formulas:

• Maximum height:

• Relationship between initial velocities:

Step-by-Step Guidance

1. Write the maximum height for each stone in terms of its initial velocity.

2. Express the faster stone's height in terms of the slower stone's height .

3. Set up the ratio and simplify.

Try solving on your own before revealing the answer!

Free Response 2(a). A dog in an open field is at rest under a tree at and starts running. Its acceleration is (m/s). Find the velocity vector .

Background

Topic: Vector Kinematics, Integration

This question tests your ability to integrate a time-dependent acceleration vector to find velocity, given initial conditions.

Key Terms and Formulas:

• Initial velocity:

Step-by-Step Guidance

1. Integrate each component of the acceleration vector with respect to time.

2. Add the initial velocity vector to your result.

3. Write the final expression for in terms of and .

Try solving on your own before revealing the answer!

Free Response 2(b). Find the position vector measured from the tree.

Background

Topic: Vector Kinematics, Integration

This question tests your ability to integrate the velocity vector to find position, given initial conditions.

Key Terms and Formulas:

• Initial position:

Step-by-Step Guidance

1. Integrate each component of the velocity vector with respect to time.

2. Add the initial position vector to your result.

3. Write the final expression for in terms of and .

Try solving on your own before revealing the answer!

Free Response 2(c). Evaluate .

Background

Topic: Vector Evaluation, Substitution

This question tests your ability to substitute a specific value of into a vector function.

Key Terms and Formulas:

• Use from the previous part.

Step-by-Step Guidance

1. Substitute into each component of .

2. Calculate the numerical values for the and components.

Try solving on your own before revealing the answer!

Free Response 2(d). How far is the dog from the tree at s? (Compute and leave your answer in radical form.)

Background

Topic: Vector Magnitude

This question tests your ability to find the magnitude of a vector given its components.

Key Terms and Formulas:

• Magnitude:

Step-by-Step Guidance

1. Use the and components from the previous answer.

2. Set up the magnitude formula using these values.

3. Simplify the expression, but leave it in radical form.

Try solving on your own before revealing the answer!

Free Response 3(a). A jet plane pulls out of a downward dive. The bottom portion of the flight path is a quarter circle of radius 300 m. Medical studies show that pilots lose consciousness if the upward acceleration exceeds 5.5g. What minimum speed must the plane have at the bottom of the dive for the pilot to black out?

Background

Topic: Circular Motion, Centripetal Acceleration

This question tests your ability to relate centripetal acceleration to speed and radius, and to apply a physical constraint (maximum acceleration).

Key Terms and Formulas:

• Centripetal acceleration:

• Maximum allowed acceleration:

Step-by-Step Guidance

1. Set to find the speed at which the pilot blacks out.

2. Rearrange the formula to solve for in terms of and .

3. Plug in the given values for and .

Try solving on your own before revealing the answer!

Free Response 3(b). Express your answer in both m/s and mph. ()

Background

Topic: Unit Conversion

This question tests your ability to convert speed from meters per second to miles per hour.

Key Terms and Formulas:

• Conversion:

Step-by-Step Guidance

1. Multiply the speed in m/s by 2.24 to get mph.

Try solving on your own before revealing the answer!