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PHY 251 Exam 1 Practice – Step-by-Step Physics Guidance

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Q1. To determine the magnitude of a vector given its components, you would need to do a calculation involving:

Background

Topic: Vectors and their Magnitude

This question tests your understanding of how to find the magnitude of a vector from its components, a fundamental skill in physics.

Key Terms and Formulas:

  • Vector components: The values along the x and y axes (or i and j unit vectors).

  • Pythagorean theorem: Used to find the magnitude of a vector.

Formula:

Step-by-Step Guidance

  1. Identify the vector's components, and .

  2. Recall that the magnitude is found using the Pythagorean theorem.

  3. Set up the formula: .

Try solving on your own before revealing the answer!

Q2. If two vectors of equal magnitude are perpendicular to each other, which of the following is equal to zero?

Background

Topic: Vector Operations – Dot and Cross Products

This question tests your knowledge of how dot and cross products behave for perpendicular vectors.

Key Terms and Formulas:

  • Dot product:

  • Cross product:

Step-by-Step Guidance

  1. Recall that perpendicular vectors have .

  2. Plug into the dot product formula: .

  3. Consider the cross product: , so the cross product is not zero.

Try solving on your own before revealing the answer!

Q3. On a plot of velocity versus time in one-dimensional motion, the slope of a line tangent to the curve is equal to the object’s:

Background

Topic: Kinematics – Graphical Interpretation

This question tests your ability to interpret velocity-time graphs and understand what the slope represents.

Key Terms and Formulas:

  • Instantaneous acceleration: The derivative of velocity with respect to time.

  • Slope:

Step-by-Step Guidance

  1. Recall that the slope of a velocity vs. time graph is .

  2. Understand that gives the instantaneous acceleration.

  3. Compare this to average acceleration, which is over an interval.

Try solving on your own before revealing the answer!

Q4. For an object tossed straight up into the air, ignoring air drag, at what point(s) in the motion is the acceleration NOT equal to 9.80 m/s² downward?

Background

Topic: Kinematics – Acceleration Due to Gravity

This question tests your understanding of how gravity acts on an object throughout its trajectory.

Key Terms and Formulas:

  • Acceleration due to gravity: downward

Step-by-Step Guidance

  1. Recall that gravity acts constantly on the object, regardless of its position.

  2. Consider the motion: on the way up, at the highest point, and on the way down.

  3. Think about whether acceleration changes at any of these points.

Try solving on your own before revealing the answer!

Q5. In one-dimensional motion, if the acceleration of an object is a linear function of time, what is the degree (highest exponent) of the equation of its position as a function of time?

Background

Topic: Kinematics – Position Functions

This question tests your understanding of how integrating acceleration affects the position equation.

Key Terms and Formulas:

  • Acceleration: (linear in time)

  • Velocity:

  • Position:

Step-by-Step Guidance

  1. Write acceleration as .

  2. Integrate to get velocity: .

  3. Integrate to get position: .

Try solving on your own before revealing the answer!

Q6. For any motion, the vector for the average acceleration over a time interval always points in the same direction as the vector for which quantity, measured over that time interval?

Background

Topic: Vectors – Acceleration and Velocity

This question tests your understanding of the relationship between acceleration and other vector quantities.

Key Terms and Formulas:

  • Average acceleration:

  • Change in velocity:

Step-by-Step Guidance

  1. Recall the formula for average acceleration.

  2. Consider the direction of .

  3. Compare with other vector quantities: position, displacement, average velocity.

Try solving on your own before revealing the answer!

Q7. For projectile motion with negligible air resistance, what property of the object’s motion remains the same throughout the flight?

Background

Topic: Projectile Motion – Conservation of Components

This question tests your understanding of which components of motion are conserved in projectile motion.

Key Terms and Formulas:

  • Horizontal velocity:

  • Vertical velocity:

Step-by-Step Guidance

  1. Recall that gravity only affects the vertical component.

  2. Think about which component is unaffected by gravity.

  3. Consider how speed and total velocity change during flight.

Try solving on your own before revealing the answer!

Q8. What direction does the acceleration vector point for an object moving in a circle and slowing down at a steady rate?

Background

Topic: Circular Motion – Acceleration Components

This question tests your understanding of tangential and centripetal acceleration in circular motion.

Key Terms and Formulas:

  • Centripetal acceleration: points toward the center.

  • Tangential acceleration: points along the tangent (direction of motion or opposite).

Step-by-Step Guidance

  1. Recall that slowing down means tangential acceleration is opposite the direction of motion.

  2. Centripetal acceleration always points toward the center.

  3. Think about the vector sum of these two accelerations.

Try solving on your own before revealing the answer!

Q9. Draw two vectors, A and B (not along the same line), and show how you would graphically construct the sum C = A + B and the difference D = A – B.

Background

Topic: Vector Addition and Subtraction – Graphical Methods

This question tests your ability to represent vector addition and subtraction graphically.

Key Terms and Formulas:

  • Vector addition: Place the tail of B at the head of A.

  • Vector subtraction: Reverse B, then add to A.

Step-by-Step Guidance

  1. Draw vector A and vector B with different directions.

  2. For C = A + B, place the tail of B at the head of A; draw the resultant from the tail of A to the head of B.

  3. For D = A – B, reverse the direction of B to get –B, then add to A as before.

Try sketching this before checking your answer!

Q10. Explain the conceptual connection between average velocity and instantaneous velocity.

Background

Topic: Kinematics – Velocity Concepts

This question tests your understanding of the difference and relationship between average and instantaneous velocity.

Key Terms and Formulas:

  • Average velocity:

  • Instantaneous velocity:

Step-by-Step Guidance

  1. Recall that average velocity is over a finite interval.

  2. Instantaneous velocity is the limit as the interval approaches zero.

  3. Think about how the instantaneous velocity relates to the slope of the position vs. time graph at a point.

Try explaining this in your own words before checking the answer!

Q11. If you toss an object straight up, there will be two times (one on the way up and one on the way down) at which it is at a given height between the launch height and the maximum height. Which of the motion equations for constant acceleration is of a form that would suggest two possible solutions for the time?

Background

Topic: Kinematics – Quadratic Equations in Motion

This question tests your understanding of how the equations of motion can yield multiple solutions for time.

Key Terms and Formulas:

  • Position equation:

Step-by-Step Guidance

  1. Write the position equation for constant acceleration.

  2. Set to the desired height and rearrange the equation.

  3. Recognize that this is a quadratic equation in .

Try identifying the equation before checking the answer!

Q12. At what point(s) along a projectile’s trajectory (ignoring air resistance) is the velocity vector perpendicular to the acceleration vector?

Background

Topic: Projectile Motion – Vector Relationships

This question tests your understanding of the relationship between velocity and acceleration vectors in projectile motion.

Key Terms and Formulas:

  • Velocity vector: Tangent to the trajectory.

  • Acceleration vector: Always downward (gravity).

Step-by-Step Guidance

  1. Recall that acceleration is always vertical downward.

  2. Think about the direction of velocity at different points along the trajectory.

  3. Identify the point where velocity is horizontal.

Try reasoning through the trajectory before checking the answer!

Q13. An object travelling along a circular path slows down to one third of its original speed. By what factor does its centripetal acceleration change? Explain how you determined this.

Background

Topic: Circular Motion – Centripetal Acceleration

This question tests your understanding of how centripetal acceleration depends on speed.

Key Terms and Formulas:

  • Centripetal acceleration:

Step-by-Step Guidance

  1. Write the formula for centripetal acceleration.

  2. Substitute the new speed () into the formula.

  3. Compare the new acceleration to the original.

Try calculating the factor before checking the answer!

Q14. Given the vectors A = 7.20i – 3.50j and B = -4.10i + 6.90j, find a) A · B and b) the angle between the vectors.

Background

Topic: Vector Algebra – Dot Product and Angle Calculation

This question tests your ability to compute the dot product and use it to find the angle between two vectors.

Key Terms and Formulas:

  • Dot product:

  • Magnitude:

  • Angle:

Step-by-Step Guidance

  1. Identify the components: , , , .

  2. Calculate the dot product: .

  3. Find the magnitudes: and .

  4. Set up the angle formula: .

Try calculating these values before checking the answer!

Q15. A car is stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by x(t) = bt – ct², where b = 8.40 m/s and c = 0.610 m/s². Calculate a) the average velocity of the car for the time interval t = 3.70 s to t = 9.20 s, and b) the instantaneous velocity of the car at t = 5.40 s.

Background

Topic: Kinematics – Position and Velocity Functions

This question tests your ability to use position functions to find average and instantaneous velocity.

Key Terms and Formulas:

  • Position function:

  • Average velocity:

  • Instantaneous velocity:

Step-by-Step Guidance

  1. For average velocity, calculate and using the position function.

  2. Plug these values into .

  3. For instantaneous velocity, differentiate : .

  4. Plug into to find the instantaneous velocity.

Try working through the calculations before checking the answer!

Q16. A certain volcano on Earth can eject rocks vertically to a maximum height of 150 m. a) How high would these rocks go if a volcano on Mars ejected them with the same initial velocity? The acceleration on Mars is 3.71 m/s². Ignore air resistance on both planets. b) How long would they be in the air on Mars?

Background

Topic: Kinematics – Projectile Motion and Gravity

This question tests your ability to apply projectile motion equations with different gravitational accelerations.

Key Terms and Formulas:

  • Maximum height:

  • Time in air:

Step-by-Step Guidance

  1. First, find the initial velocity on Earth using .

  2. Use this to find the maximum height on Mars: .

  3. For time in air on Mars, use .

Try setting up the equations before checking the answer!

Q17. Firemen use a high-pressure hose to shoot a stream of water at a burning building. The water has a speed of 27.0 m/s as it leaves the end of the hose and then exhibits projectile motion. The firemen adjust the angle of elevation θ of the hose until the water takes 3.40 s to reach a building 48.0 m away. Ignore air resistance, and assume that the end of the hose is at ground level. a) Find θ. b) How high above the ground does the water strike the building?

Background

Topic: Projectile Motion – Time, Range, and Height

This question tests your ability to analyze projectile motion with given speed, time, and distance.

Key Terms and Formulas:

  • Horizontal distance:

  • Vertical position:

Step-by-Step Guidance

  1. Set up the horizontal equation: .

  2. Solve for and then .

  3. Use in the vertical equation to find at .

Try solving for θ and y before checking the answer!

Q18. A Ferris wheel with radius 19.0 m is turning about a horizontal axis through its center. The linear speed of a passenger on the rim is constant, and the period of her circular motion is 16.0 s. What are the magnitude and direction of the passenger’s acceleration as she passes through a) the lowest point in her circular motion and b) the highest point in her circular motion?

Background

Topic: Circular Motion – Centripetal Acceleration

This question tests your ability to calculate centripetal acceleration and understand its direction at different points in a circle.

Key Terms and Formulas:

  • Centripetal acceleration:

  • Linear speed:

Step-by-Step Guidance

  1. Calculate the linear speed: .

  2. Use to find the magnitude of acceleration.

  3. At the lowest point, the acceleration points upward toward the center; at the highest point, it points downward toward the center.

Try calculating the values and directions before checking the answer!

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