Short answers to common questions.
1) Why are horizontal and vertical projectile motions independent?
Because gravity only acts vertically — a_x = 0. The component-independence principle of §4.2: if acceleration is only along one axis, motion along the others is uniform.
2) If I drop and throw (horizontally) two balls from the same window, which lands first?
Simultaneously. Horizontal velocity doesn't affect vertical fall. Galileo's beautiful discovery.
3) Why isn't 45° always the optimal launch angle?
Only if launch height = landing height. From higher ground, \theta^* < 45°. With air drag, usually 35°-40°.
4) Is centripetal acceleration real? Or a force?
The acceleration is real, but it isn't a force by itself — it's the result of a force applied toward the center (rope, gravity, friction, ...). That force is called "centripetal force".
5) What is "centrifugal force"?
Not real — an apparent effect in a rotating frame. From outside (inertial), only centripetal force exists. From inside the rotating frame, you feel a "fictitious force" outward.
6) If a car speeds up in a curve, what happens?
a_c = v^2/r — increases with the square of speed. For the same curve radius, you need more friction. If friction can't provide it, the car slips.
7) Why do ISS astronauts appear weightless?
Not because gravity is zero — g is still ~8.7. They're in free fall. The station accelerates downward under gravity, but its horizontal speed keeps it missing the surface. That's an orbit.
8) How do I tell if a frame is inertial?
Release an object. If it stays put or moves in a straight line at constant velocity, inertial. If it accelerates with no apparent force, non-inertial.
9) In horizontal launch, when does the ball momentarily stop?
Never — horizontal launch has constant horizontal velocity until impact. Vertical velocity grows from zero and never returns to zero (until it hits).
10) Is projectile acceleration zero at the peak?
No. At the peak, v_y = 0 but \vec a = -g\hat j still. That's why it immediately starts falling.
11) If there were no air, what would change?
- Cleaner soccer trajectories (no drag, no Magnus effect)
- Feather and hammer fall together (Apollo 15)
- Infinite speed possible (in dreams) — but no breathing
12) How do I tell if a problem is 2D or 3D?
Look at the motion directions. Only one line — 1D. Confined to a plane — 2D. General space — 3D. Projectiles are usually 2D (vertical plane); satellite circular orbit is 2D.
13) In circular motion, angular velocity vs linear speed?
- Linear speed
v: m/s — how fast along the path - Angular velocity
\omega: rad/s — how fast the angle changes - Relation:
v = \omega r
14) If a projectile returns to launch height, is impact speed = launch speed?
Yes. Symmetry. The angle mirrors as well.
15) In relative velocity, does order matter?
Yes. \vec v_{A/B} = -\vec v_{B/A}. Your velocity relative to the car ahead is the negative of its relative to you.
16) If I throw a ball straight up inside a train, does it land in the same place?
Yes — provided the train has constant velocity. Inside, you see straight up-and-down. From ground, a parabola.
17) What does a projectile look like without gravity?
Straight line at constant velocity (Newton's first). No force, no acceleration.
18) Which formulas should I always start from for projectile problems?
Always: x(t) = v_{0x} t and y(t) = v_{0y} t - \frac{1}{2}g t^2. Everything else (peak, range) follows.
19) In a rotating frame, do Newton's laws hold as-is?
No. Rotating frames are non-inertial. To preserve F = ma, we add "fictitious forces" (centrifugal, Coriolis). Deeper in Ch 10-11.
20) What's really new here beyond Chapter 2?
Chapter 2 was 1D with +/- signs. Chapter 4 lifts everything to 2D/3D with vector tools. Same principles (velocity is derivative of position, acceleration is derivative of velocity), now in real space.
Chapter wrap-up
Three main ideas:
- Principle of component independence — 2D/3D kinematics decomposes into simple 1D motions
- Projectile = uniform horizontal + free-fall vertical — two independent motions on one particle
- Circular motion has acceleration — even at constant speed — because direction changes
Chapter 5 — Newton's laws: why these motions happen.
🎓 You've finished Chapter 4!
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