2025.02

Homework Guidelines

• Students should submit homework in a vertical format, step-by-step down the left of the page.

• Equations must be complete and clearly explain the calculations being performed. For example, instead of just stating the outcome such as (\theta = \tan^{-1}(x) = 63^\circ), provide context, including previous values calculated or defined in the problem (e.g., where (x = 10N) and (y = 20N)).

Concepts in Momentum Problems

• Conservation of Momentum:

  • Momentum before a collision equals momentum after, denoted as: (Δ(Σp) = 0).

• Momentum and Force Relationship:

  • Using the momentum version of Newton’s 2nd Law: (FΔt = Δ(mv)).

Example Problem: Colliding Balls

• Question: If a ping pong ball (2.5 g) and a bowling ball (6 kg) collide, which ball's momentum changes by the largest magnitude?

  • Options:

    • A: The ping pong ball.

    • B: The bowling ball.

    • C: It depends on the balls’ initial velocities.

    • D: Both momenta change by the same amount.

• Velocity Change Consideration:

  • If the same two balls are analyzed for velocity change, the question must account for initial velocities in the same manner.

Clay Momentum Example

• A lump of 5 kg clay falls and sticks to the ground. What occurs to its momentum?

  • Options include:

    • A: Momentum is conserved (false).

    • B: It turns into heat (false).

    • C: It was transferred to the Earth (true).

    • D: It went into the air (false).

Additional Example Problems

• Quarterback vs. Lineman: 85 kg quarterback moves at 2.1 m/s, and a 125 kg lineman attempts to stop him. Calculate the lineman’s speed needed in the opposite direction to stop the quarterback.

• Bullet Acceleration Problem: A bullet weighing 11.7 g accelerated in a gun barrel of length 0.51 m with an acceleration of (3.8 × 10^5 \text{ m/s}^2). Solve for:

  • a) Force on the bullet

  • b) Time in the barrel

  • c) How long the bullet pushes on the gun

  • d) Bullet's momentum as it exits the gun.

Momentum Change Representation

• Bigger changes in momentum correspond to different scenarios:

  • A: Ball bounces off a wall.

  • B: Ball sticks to a wall that is glued.

  • C: Both cases represent the same change in momentum.

Velocity Change Problems

• 1 kg ball at +2 m/s:

  • Hits a wall and sticks: Change in velocity is:

    • A: 0 m/s

    • B: +2 m/s

    • C: −2 m/s (Correct)

    • D: +4 m/s

    • E: −4 m/s

  • Hits a wall and bounces back at −2 m/s: Change is:

    • A: 0 m/s

    • B: +2 m/s

    • C: −2 m/s

    • D: +4 m/s

    • E: −4 m/s (Correct)

Motion on Curved Paths

• Key Reminders:

  • Keep track of various velocities with subscripts; different forces will affect motion.

  • Acceleration and velocity are distinct entities.

  • Understand laws of motion connecting forces with acceleration, velocity, displacement, and time.

  • Recognize angular quantities resembling linear motion: angular displacement, velocity, acceleration, and momentum.

General Concepts

a. Horizontal Motion with Vertical Drop:

• Upon throwing a rock horizontally off a cliff: It will take the same time to hit the ground as a rock simply dropped from the same height, due to gravity acting equally on both.

Invalid Calculations & Misconceptions

• Common mistakes include faulty calculations for elapsed time of free-falling objects:

  • Example: A horizontal toss from a 5 m height at 2.5 m/s misjudged as (t = \frac{5.0 m}{2.5 m/s} = 2 s) disregards vertical acceleration from gravity.

Discussion Points & Applications

• Groups discuss scenarios like shells fired from a battleship and the timing of projectile motion under gravity.

• Visualize trajectories and parabolic motions for object movements in gravitational fields, considering different orbits caused by varying initial velocities.