Q1. A car is moving at a speed of 150 km/h. Express this speed in meters per second and in miles per hour. (1 mi = 1.609 km)

Background

Topic: Unit Conversion

This question tests your ability to convert between different units of speed, specifically from kilometers per hour (km/h) to meters per second (m/s) and miles per hour (mi/h).

Key Terms and Formulas

• 1 km = 1000 m

• 1 hour = 3600 seconds

• 1 mile = 1.609 km

Step-by-Step Guidance

1. Start by converting 150 km/h to meters per second. Set up the conversion using the relationships above.

3. Multiply out the numerators and denominators, but do not calculate the final value yet.

4. For miles per hour, use the conversion:

5. Set up the multiplication, but stop before calculating the final result.

Try solving on your own before revealing the answer!

Q2. (a) What is the age of the universe (13.7 billion years) in seconds? (b) Our species evolved about seconds ago. How many years is that?

Background

Topic: Scientific Notation and Unit Conversion

This question tests your ability to convert between years and seconds using scientific notation and to interpret large numbers in different units.

Key Terms and Formulas

• 1 year = 365.25 days (accounting for leap years)

• 1 day = 24 hours

• 1 hour = 3600 seconds

• Scientific notation for large numbers

Step-by-Step Guidance

1. For part (a), write 13.7 billion years in scientific notation: years.

2. Set up the conversion:

3. Multiply the factors, but do not compute the final value yet.

4. For part (b), set up the reverse conversion:

5. Multiply the factors, but stop before the final calculation.

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Q3. The strength of the force (F) is related to the speed of the object by , where b is a constant. In SI units, the units of force are kg·m/s² and the units of speed are m/s. For the equation to have consistent units, what are the units of b in SI?

Background

Topic: Dimensional Analysis

This question tests your understanding of dimensional consistency and how to determine the units of a constant in a physical equation.

Key Terms and Formulas

• Force (): SI unit is kg·m/s²

• Speed (): SI unit is m/s

• Dimensional consistency: Both sides of an equation must have the same units.

Step-by-Step Guidance

1. Write the units for each term: ,

2. Substitute the units into the equation:

3. Express in terms of SI units:

4. Set up the equation for units:

5. Rearrange to solve for , but do not compute the final units yet.

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Q4. Calculate the value of energy (E) for an electron, using powers of 10 and three significant figures. , where m/s and kg.

Background

Topic: Significant Figures and Scientific Notation

This question tests your ability to use scientific notation and significant figures in calculations involving physical constants.

Key Terms and Formulas

• Energy-mass equivalence:

• Speed of light (): m/s

• Electron mass (): kg

• Significant figures: Use three significant digits in your answer.

Step-by-Step Guidance

1. Write the equation:

2. Substitute the given values:

3. Calculate first:

4. Multiply the result by , keeping track of powers of 10 and significant figures.

5. Stop before writing the final value for .

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Q5. Find the results: and

Background

Topic: Significant Figures and Order of Operations

This question tests your ability to perform calculations with the correct number of significant figures and to follow the order of operations.

Key Terms and Formulas

• Significant figures: The number of digits that carry meaning in a measurement.

• Order of operations: Parentheses, multiplication/division, addition/subtraction.

Step-by-Step Guidance

1. For the division, first multiply .

2. Divide by the result from step 1.

3. For the addition, add .

4. Round both results to the correct number of significant figures (based on the rules for multiplication/division and addition/subtraction).

5. Stop before stating the final rounded values.

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Q6. You travel 2.00 km north and then 5.00 km west. How far and in what direction are you from your starting point? (What are the magnitude and direction of your resultant displacement?)

Background

Topic: Vector Addition (Displacement)

This question tests your ability to add vectors using the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant displacement.

Key Terms and Formulas

• Displacement: A vector quantity with both magnitude and direction.

• Pythagorean theorem:

• Direction (angle):

Step-by-Step Guidance

1. Draw a right triangle with legs of 2.00 km (north) and 5.00 km (west).

2. Use the Pythagorean theorem to set up the calculation for the magnitude of the displacement:

3. Set up the calculation for the direction: , where is measured north of west.

4. Stop before calculating the final magnitude and angle.

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Q7. A dog walks a certain distance straight east and then a certain distance straight north. It ends up at a position that is 600 m from the house in a direction that is 25° north of east. A cat, instead, walks first straight west and then straight south. Its final position is 800 m from the starting point in a direction that is 65° south of west. Assume that the house is at the origin. How far did the dog walk on the east leg of its trip? How far on the north leg? Similarly, how far did the cat walk on the west leg and how far on the south leg?

Background

Topic: Vector Components

This question tests your ability to resolve a vector into its components using trigonometry, given the magnitude and direction.

Key Terms and Formulas

• Vector components: ,

• Direction: Angle measured from the positive x-axis (east for the dog, west for the cat)

Step-by-Step Guidance

1. For the dog: Use m and north of east.

2. Set up the east (x) component:

3. Set up the north (y) component:

4. For the cat: Use m and south of west.

5. Set up the west (x) component: (negative x-direction)

6. Set up the south (y) component: (negative y-direction)

7. Stop before calculating the final values for each leg.

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Q8. Vector A has a magnitude of 60 cm and a direction of 40°, and vector B has a magnitude of 45 cm and a direction of 120°. Both angles are measured counterclockwise from the positive axis. Use components to calculate the magnitude and direction of the vector sum.

Background

Topic: Vector Addition Using Components

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

Key Terms and Formulas

• Component form: ,

• Resultant vector: ,

• Magnitude:

• Direction:

Step-by-Step Guidance

1. Find the x and y components of vector A: ,

2. Find the x and y components of vector B: ,

3. Add the x components:

4. Add the y components:

5. Set up the magnitude and direction formulas, but stop before calculating the final values.

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Q9. Four students are participating in a campus scavenger hunt. They all start at the same location and receive the following three displacements (in a different order for each student): 52.0 m, 30.0° east of north; 41.0 m, 45.0° south of east; 35.0 m west. What does the winner calculate for the final location?

Background

Topic: Vector Addition (Multiple Displacements)

This question tests your ability to add multiple vectors with different directions using components to find the final position.

Key Terms and Formulas

• Component form: ,

• Sum all x-components and y-components separately

• Magnitude:

• Direction:

Step-by-Step Guidance

1. Break each displacement into x and y components:

   • 52.0 m at 30.0° east of north

   • 41.0 m at 45.0° south of east

   • 35.0 m west (negative x-direction)

2. For each vector, use and as appropriate for the given angles.

3. Add all x-components to get and all y-components to get .

4. Set up the magnitude and direction formulas for the resultant vector, but stop before calculating the final values.

Try solving on your own before revealing the answer!