Q1. Which one of these is not a vector?

Background

Topic: Scalars vs. Vectors

This question tests your understanding of the difference between scalar and vector quantities in physics. Vectors have both magnitude and direction, while scalars have only magnitude.

Key Terms:

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

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

Step-by-Step Guidance

1. Review the definitions of each option: displacement, distance, acceleration, and velocity.

2. Identify which of these quantities does not require a direction to be fully described.

3. Recall that vectors are typically represented with arrows and can be positive or negative depending on direction, while scalars are always positive.

Try solving on your own before revealing the answer!

Q2. Relative velocity: Leo running on a train

Background

Topic: Relative Motion in One Dimension

This question tests your ability to use the concept of relative velocity to relate the velocities of objects as measured from different reference frames.

Key Formula:

• : Velocity of A relative to B

• : Velocity of A relative to C

• : Velocity of B relative to C

Step-by-Step Guidance

1. Define the reference frames: Leo (A), Train (B), Director (C).

2. Write the equation for Leo's velocity relative to the train:

3. Plug in the given values: m/s (east), m/s (east).

4. Set up the subtraction, being careful with the direction (signs) of each velocity.

Try solving on your own before revealing the answer!

Q3. Position vs. Time Graph: Signs of Acceleration and Velocity at Point P

Background

Topic: Kinematics Graphs

This question tests your ability to interpret position vs. time graphs and determine the signs of velocity and acceleration at a specific point.

Key Concepts:

• Velocity: The slope of the position vs. time graph at a point.

• Acceleration: The curvature (concavity) of the graph; if the graph is concave up, acceleration is positive; if concave down, acceleration is negative.

Step-by-Step Guidance

1. Look at the slope of the curve at point P to determine the sign of the velocity.

2. Examine the curvature (is the graph bending upwards or downwards?) at point P to determine the sign of the acceleration.

3. Recall: If the graph is curving downward (concave down), acceleration is negative; if curving upward (concave up), acceleration is positive.

Try solving on your own before revealing the answer!

Q4. Cost of driving to grandma’s house in dollars per second

Background

Topic: Unit Conversion and Rates

This question tests your ability to use unit analysis and rates to convert the cost of a trip into dollars per second.

Key Steps and Formulas:

• Find the total gallons used:

• Find the total cost:

• Find the total time:

• Convert time to seconds if needed.

• Calculate cost per second:

Step-by-Step Guidance

1. Calculate the number of gallons needed for a 40-mile trip with 30 miles/gallon efficiency.

2. Multiply the gallons by the cost per gallon to get the total cost.

3. Calculate the time taken for the trip using the average speed (convert hours to seconds).

4. Set up the division to find the cost per second.

Try solving on your own before revealing the answer!

Q5. Sprinter’s total distance from A to B (acceleration, constant speed, deceleration)

Background

Topic: Kinematics with Constant Acceleration

This question tests your ability to break a motion into segments (acceleration, constant speed, deceleration) and use kinematic equations to find total distance.

Key Formulas:

• For acceleration:

• For constant speed:

• For deceleration: (solve for )

Step-by-Step Guidance

1. Calculate the distance covered during the acceleration phase (first 5.0 s, starting from rest).

2. Find the velocity at the end of the acceleration phase to use for the constant speed segment.

3. Calculate the distance covered during the constant speed phase (15 s).

4. Use the deceleration rate and final velocity (0 m/s) to find the distance during the stopping phase.

5. Add up the distances from all three segments to get the total distance.

Try solving on your own before revealing the answer!

Q6. Magnitude of the hobbits’ total displacement (vector addition)

Background

Topic: Vector Addition and Displacement

This question tests your ability to add vectors (using the law of cosines or components) to find the magnitude of the total displacement.

Key Formula:

If two vectors and are at an angle to each other:

Step-by-Step Guidance

1. Draw a vector diagram showing the two legs of the journey (650 mi east, 1258 mi southeast).

2. Determine the angle between the two vectors (southeast is 45° from east).

3. Set up the law of cosines with the given magnitudes and angle.

4. Plug in the values and prepare to compute the magnitude of the resultant displacement.

Try solving on your own before revealing the answer!

Q7a. Draw a diagram of projectile motion over the city walls

Background

Topic: Projectile Motion Diagrams

This part asks you to visually represent the projectile motion of a stone, labeling velocity and acceleration vectors at key points.

Key Concepts:

• At launch: Velocity has both horizontal and vertical components; acceleration is downward (gravity).

• At the peak: Velocity is purely horizontal; acceleration is still downward.

• Halfway up: Both velocity components are present; acceleration is downward.

Step-by-Step Guidance

1. Draw a parabolic trajectory starting from the catapult and ending just over the wall.

2. At the launch point, draw and label the velocity vector at a 45° angle and the acceleration vector straight down.

3. At the peak, draw the velocity vector horizontally and the acceleration vector downward.

4. Halfway up, show both velocity components and the downward acceleration.

Try sketching your own diagram before revealing the answer!

Q7b. Calculate the minimum initial speed to clear the wall

Background

Topic: Projectile Motion Calculations

This question tests your ability to use projectile motion equations to find the minimum initial speed needed to clear a wall at a given distance and height.

Key Formulas:

• Horizontal motion:

• Vertical motion:

• Given: m, m,

Step-by-Step Guidance

1. Express time in terms of using the horizontal motion equation.

2. Substitute this expression for into the vertical motion equation.

3. Set m and m, and solve for .

4. Set up the resulting equation for and prepare to solve for its value.

Try solving on your own before revealing the answer!

Q8. Baseball mitt thrown vertically to intercept a home run

Background

Topic: Relative Motion and Projectile Interception

This question tests your ability to analyze projectile motion and determine the initial speed needed for one object (the mitt) to intercept another (the ball) at the same height and time.

Key Steps and Formulas:

• Find the time it takes for the ball to reach the position above you (25 m away).

• Use the vertical motion equation for the mitt:

• Set equal to the height of the ball when it is above you, and solve for .

Step-by-Step Guidance

1. Calculate the time it takes for the ball to travel 25 m horizontally using its horizontal velocity component.

2. Find the height of the ball at that time using the vertical motion equation.

3. Set up the equation for the mitt to reach that height in the same time interval.

4. Prepare to solve for the required initial speed of the mitt.

Try solving on your own before revealing the answer!