Physics I: Mechanics – Sample Exam Problems and Grading Rubric

Grading Rubric for Physics Problems

This rubric outlines the criteria used to grade physics exam questions, focusing on the correctness of solutions and the clarity of work shown.

Score	Description
100%	Solution is correct and work is clearly shown.
80%	One or more minor mistakes have been made. Perhaps there is an arithmetic error or a unit conversion was missed. Work is clearly shown. (Minor mistakes that do not affect the final answer may be overlooked at the grader's discretion.)
60%	There are one or more significant mistakes, but the overall approach to the problem is correct. Perhaps the solution is setup wrong, or an equation was used incorrectly. Work is shown, but may be difficult to follow.
40%	The overall approach to the problem is incorrect, but at least some portion of the solution is correct.
0%	Solution is correct but work is not shown. OR The solution is wrong or missing.

Problem 1: Dynamics of Connected Blocks on an Inclined Plane

System Description

Two blocks, Block 1 and Block 2, each with mass $m_1 = m_2 = 4.0\ \mathrm{kg}$, are connected by a rope. Block 2 rests on a frictionless ramp inclined at $\theta = 20^\circ$, while Block 1 hangs vertically off the edge of the ramp. The system is illustrated in the provided diagram.

• Key Concepts: Newton's Second Law, free-body diagrams, normal force, tension, acceleration.

• Assumptions: The ramp is frictionless; the rope is massless and inextensible.

Subtopics and Sample Solutions

Normal Force on Block 2

• The normal force is the perpendicular contact force exerted by the ramp on Block 2.

• To find the normal force $N$:

$N = m_2 g \cos\theta$

• Example: For $m_2 = 4.0\ \mathrm{kg}$ and $\theta = 20^\circ$:

$N = (4.0\ \mathrm{kg})(9.8\ \mathrm{m/s^2})\cos(20^\circ) = 36.8\ \mathrm{N}$

Acceleration of the Blocks

• Apply Newton's Second Law to each block.

• Set up equations for forces parallel to the ramp and vertically for Block 1.

• Determine if the system accelerates and in which direction.

$T - m_1 g = -m_1 a$ $m_2 g \sin\theta - T = m_2 a$

• Solve for $a$ and determine its sign to see if Block 1 moves up or down.

Tension in the Rope

• Once $a$ is found, substitute back to find the tension $T$ in the rope.

$T = m_1 (g - a)$

Problem 2: Projectile Motion – Soccer Ball Kicked Toward a Lamppost

System Description

A soccer ball is kicked with an initial speed of $24\ \mathrm{m/s}$ at an angle $\theta = 25^\circ$ above the horizontal. The ball hits a lamppost at a height of $3.0\ \mathrm{m}$ above the ground. The problem involves analyzing the projectile's motion to determine its maximum height, speed at impact, and horizontal distance traveled.

• Key Concepts: Kinematics of projectile motion, vector decomposition, conservation of energy.

Subtopics and Sample Solutions

Maximum Height

• Vertical component of initial velocity: $v_{0y} = v_0 \sin\theta$

• Maximum height $h_{max}$ above the ground:

$h_{max} = \frac{v_{0y}^2}{2g}$

• Add the initial height if not starting from ground level (not needed here).

Speed at Impact

• Find the $x$ and $y$ components of velocity just before impact.

• Use kinematic equations to find $v_y$ at $y = 3.0\ \mathrm{m}$.

• Total speed:

$v = \sqrt{v_x^2 + v_y^2}$

Horizontal Distance

• Find the time $t$ when the ball reaches $y = 3.0\ \mathrm{m}$.

• Horizontal distance $x = v_{0x} t$.

$v_{0x} = v_0 \cos\theta$ x = $v_{0x} t$

Problem 3: Vector Displacement in Three Dimensions

System Description

Simon dives underwater, moving in three straight segments at constant depth. The problem involves expressing vectors in unit vector notation, converting to magnitude and direction, and finding total displacement and distance.

• Key Concepts: Vector addition, unit vector notation, magnitude and direction, displacement vs. distance.

Subtopics and Sample Solutions

Unit Vector Notation

• Express a vector $\vec{r}$ as $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.

• Convert between magnitude/direction and components using trigonometry.

Total Displacement

• Add all displacement vectors to find total displacement:

$\vec{R}_{\text{total}} = \vec{r}_1 + \vec{r}_2 + \vec{r}_3$

• Magnitude of displacement:

$|\vec{R}_{\text{total}}| = \sqrt{(x_{\text{total}})^2 + (y_{\text{total}})^2 + (z_{\text{total}})^2}$

Distance Traveled

• Total distance is the sum of the magnitudes of each segment:

$\text{Distance} = |\vec{r}_1| + |\vec{r}_2| + |\vec{r}_3|$

Example Table: Comparison of Displacement and Distance

Quantity	Definition	Calculation
Displacement	Straight-line vector from start to end	Vector sum of all segments
Distance	Total path length traveled	Sum of magnitudes of all segments

Additional info: The problems are representative of introductory college-level physics, focusing on Newtonian mechanics, kinematics, and vector analysis. These are typical exam or homework questions designed to test conceptual understanding and problem-solving skills.