Motion with Constant Acceleration
High-Yield Summary
- Four kinematic equations relate v₀, v, a, t, and Δx for any constant-acceleration scenario — pick the one missing the variable you don't have.
- Free fall: Δx = ½gt², with g ≈ 9.81 m/s²; in a vacuum all objects fall at the same rate regardless of mass.
- Projectile motion splits into independent horizontal (constant velocity) and vertical (constant downward acceleration) components.
- On an inclined plane, gravity splits into a component parallel to the surface (mg sin θ) and one perpendicular to it (mg cos θ).
- Uniform circular motion requires centripetal force (F = mv²/r) directed toward the center; centrifugal force is not real — it's a fictitious effect of inertia.
The Four Kinematic Equations
v = v₀ + at | Δx = v₀t + ½at² | v² = v₀² + 2aΔx | Δx = ½(v₀ + v)t
- v₀ = Initial velocity
- v = Final velocity
- a = Acceleration (constant)
- t = Time
- Δx = Displacement
- v = v₀ + at doesn't need Δx.
- Δx = v₀t + ½at² doesn't need v.
- v² = v₀² + 2aΔx doesn't need t.
- Δx = ½(v₀ + v)t doesn't need a.
Projectile Motion: Horizontal vs. Vertical
| Horizontal Motion | Vertical Motion |
|---|---|
| No acceleration (constant velocity) | Constant, g ≈ 9.81 m/s² downward |
| v0x = v0 cos θ | v0y = v0 sin θ |
| Unchanged at the peak | Zero at the peak |
Projectile Range and Centripetal Force
R = (v0² sin 2θ) / g | F = mv² / r
- R = Horizontal range of a projectile
- θ = Launch angle
- F = Centripetal force, directed toward the center of the circle
- r = Radius of the circular path
Common MCAT Trap
- Centrifugal force isn't a real force — the outward feeling in circular motion is inertia resisting the inward pull of the real centripetal force. If a car skids outward on a turn, it's from insufficient centripetal force, not an outward force.
- On an incline, don't apply mg straight down when resolving forces — split it into mg sin θ (parallel) and mg cos θ (perpendicular) relative to the incline surface.
Quick Recall
Which kinematic equation would you use if you know v₀, a, and Δx but not t?
Why do horizontal and vertical motion stay independent in projectile motion?
Is centrifugal force real?