Physics Study Guide: Vectors, Kinematics, and Projectile Motion

Q1. If the sun casts a shadow on a 50 ft building that is 100 ft long, at what angle above the horizon is the sun?

Background

Topic: Trigonometry in Physics (Shadows and Angles)

This question tests your ability to relate the height of an object and the length of its shadow to the angle of elevation of the sun using trigonometric relationships.

Key Terms and Formulas

• Angle of elevation (θ): The angle between the ground and the line from the tip of the shadow to the top of the object.

• Tangent function: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Step-by-Step Guidance

1. Identify the height of the building (opposite side) as 50 ft and the length of the shadow (adjacent side) as 100 ft.

2. Set up the tangent relationship: $\tan \theta = \frac{50}{100}$

3. To find the angle, use the inverse tangent function: $\theta = \tan^{-1} \left( \frac{50}{100} \right)$

Try solving on your own before revealing the answer!

Q2. Find A+B and A−B if A = (28 ft, 48°) and B = (28 m, 18°)

Background

Topic: Vector Addition and Subtraction (Polar to Cartesian Conversion)

This question tests your ability to add and subtract vectors given in magnitude and direction (polar form). Note: Units must be consistent before adding or subtracting vectors.

Key Terms and Formulas

• Vector components: $A_x = A \cos \theta_A$, $A_y = A \sin \theta_A$

• Vector addition: $\vec{C} = \vec{A} + \vec{B}$ (add x and y components separately)

• Magnitude: $|\vec{C}| = \sqrt{C_x^2 + C_y^2}$

• Direction: $\theta_C = \tan^{-1} \left( \frac{C_y}{C_x} \right)$

Step-by-Step Guidance

1. First, convert all units to be consistent (e.g., convert 28 m to feet or vice versa).

2. Find the x and y components of vector A: $A_x = 28 \cos 48^\circ$, $A_y = 28 \sin 48^\circ$

3. Find the x and y components of vector B: $B_x = 28 \cos 18^\circ$, $B_y = 28 \sin 18^\circ$

4. Add (or subtract) the corresponding components to find $(A+B)$ and $(A-B)$.

5. Set up the formulas for the magnitude and direction of the resulting vectors, but do not calculate the final values yet.

Try solving on your own before revealing the answer!

Q3a. If you drove east at 30 mph for 2 hours and then north at 30 m/s for another 2 hours, what is your final displacement from your starting position?

Background

Topic: Kinematics and Vector Displacement

This question tests your understanding of displacement as a vector and how to combine displacements in perpendicular directions.

Key Terms and Formulas

• Displacement: $\vec{d} = \vec{d}_1 + \vec{d}_2$

• Magnitude: $|\vec{d}| = \sqrt{d_x^2 + d_y^2}$

• Direction: $\theta = \tan^{-1} \left( \frac{d_y}{d_x} \right)$

Step-by-Step Guidance

1. Calculate the eastward displacement: $d_x = 30 \text{ mph} \times 2 \text{ hr}$ (convert units if necessary).

2. Calculate the northward displacement: $d_y = 30 \text{ m/s} \times 2 \text{ hr}$ (convert units if necessary).

3. Express both displacements in the same units (e.g., meters or miles).

4. Set up the formula for the magnitude and direction of the total displacement vector.

Try solving on your own before revealing the answer!

Q3b. If you started from rest and it only takes 1 min to accelerate between those speeds, what was your acceleration?

Background

Topic: Kinematics (Acceleration)

This question tests your ability to calculate acceleration given an initial and final velocity and the time taken to change between them.

Key Terms and Formulas

• Acceleration: $a = \frac{\Delta v}{\Delta t}$

• Initial velocity ($v_0$): The starting speed (convert to consistent units).

• Final velocity ($v_f$): The ending speed (convert to consistent units).

Step-by-Step Guidance

1. Convert both initial and final velocities to the same units (e.g., m/s).

2. Calculate the change in velocity: $\Delta v = v_f - v_0$

3. Convert the time interval to seconds: $\Delta t = 1 \text{ min} = 60 \text{ s}$

4. Set up the acceleration formula: $a = \frac{\Delta v}{\Delta t}$

Try solving on your own before revealing the answer!

Q4. A projectile is launched at 23 m/s at an angle of 72°; find $y_{max}$, $t_f$, and $\Delta x$.

Background

Topic: Projectile Motion

This question tests your understanding of projectile motion, including how to find maximum height, time of flight, and horizontal range.

Key Terms and Formulas

• Initial velocity components: $v_{0x} = v_0 \cos \theta$, $v_{0y} = v_0 \sin \theta$

• Maximum height: $y_{max} = \frac{v_{0y}^2}{2g}$

• Time of flight: $t_f = \frac{2 v_{0y}}{g}$

• Horizontal range: $\Delta x = v_{0x} \times t_f$

• Acceleration due to gravity: $g = 9.8 \text{ m/s}^2$

Step-by-Step Guidance

1. Calculate the initial velocity components: $v_{0x} = 23 \cos 72^\circ$, $v_{0y} = 23 \sin 72^\circ$

2. Set up the formula for maximum height: $y_{max} = \frac{v_{0y}^2}{2g}$

3. Set up the formula for time of flight: $t_f = \frac{2 v_{0y}}{g}$

4. Set up the formula for horizontal range: $\Delta x = v_{0x} \times t_f$

Try solving on your own before revealing the answer!