Motion with Constant Acceleration
Motion with constant acceleration covers the kinematic equations, free fall, projectile motion, inclined planes, and uniform circular motion.
Motion with constant acceleration describes a situation where an object's acceleration stays the same over time — its velocity changes at a steady rate, either speeding up or slowing down uniformly. This subtopic covers five related scenarios: linear motion, free fall, projectile motion, inclined planes, and uniform circular motion.
Key Takeaways
The four kinematic equations relate v₀, v, a, t, and Δx for any constant-acceleration scenario.
Free fall is motion under gravity alone: Δx = ½gt², with g ≈ 9.81 m/s².
Projectile motion splits into independent horizontal (vx = v0 cos θ, constant) and vertical (vy = v0 sin θ, decelerating then accelerating) components; range = (v0² sin 2θ)/g.
On an inclined plane, gravity splits into a parallel component (mg sin θ) and a perpendicular component (mg cos θ).
Uniform circular motion requires a centripetal force (F = mv²/r) directed toward the circle's center; centrifugal force is not real — it's a fictitious effect of inertia.
Linear Motion and the Kinematic Equations
In linear motion, an object's velocity and acceleration lie along the line of motion, so its path continues along a straight line. This doesn't have to mean strictly vertical or horizontal — the inclined surface of a ramp, for instance, also provides a path for linear motion at some angle.
Motion with constant acceleration is described by four standard kinematic equations. Each is most useful depending on which variables are known and which are unknown:
Equation | Solves For | Variable Not Needed |
|---|---|---|
v = v₀ + at | Final velocity | Displacement (Δx) |
Δx = v₀t + ½at² | Displacement | Final velocity (v) |
v² = v₀² + 2aΔx | Final velocity (no time needed) | Time (t) |
Δx = ½(v₀ + v)t | Displacement (using average velocity) | Acceleration (a) |
where v₀ is initial velocity, v is final velocity, a is acceleration, t is time, and Δx is displacement.
Free Fall
Free fall is a special case of linear motion in which an object moves solely under the influence of gravity, with no other forces (like air resistance) acting on it. In a vacuum, all objects — regardless of mass — fall at the same rate when only gravity acts on them.
The acceleration experienced during free fall is constant, denoted g, and is approximately 9.81 m/s² on Earth — meaning the object's velocity increases by 9.81 m/s with every second that passes.
Acceleration: constant, g ≈ 9.81 m/s² downward.
Velocity: increases linearly with time.
Distance: increases with the square of time, given by:
Δx = ½gt²
where Δx is the distance fallen, g is the acceleration due to gravity, and t is the time spent in free fall.
Real-world examples: a skydiver experiences free fall immediately after jumping, accelerating until reaching terminal velocity (where air resistance balances gravity). The steep drops of a roller coaster also approximate free fall, especially on near-vertical slopes.
MCAT Callout — General Problem-Solving Technique for Free Fall and Linear Motion: (1) Read and reread the problem to make sure you understand the scenario and what's being asked. (2) Identify the object(s) and time interval you're analyzing. (3) Draw a diagram — even a simple sketch clarifies the setup. (4) List known and unknown quantities to organize the information. (5) Select the appropriate kinematic equation(s) based on which variables you know and need to find. (6) Perform the calculation and double-check your work, including units.
Projectile Motion
Projectile motion occurs when an object is launched into the air and moves along a curved path under gravity. Unlike linear motion, projectile motion involves two dimensions — horizontal (x) and vertical (y) — and these components are independent of each other (the horizontal motion doesn't affect the vertical, and vice versa), even though they happen simultaneously.
Horizontal Motion
The horizontal component is uniform — no acceleration, assuming negligible air resistance. The only force involved is the one imparted at launch, which sets the initial horizontal velocity. Since no additional horizontal forces act during flight, horizontal velocity stays constant:
vx = v0x = v0 cos θ
Vertical Motion
The vertical component is governed by gravity, which causes a constant downward acceleration (g ≈ 9.81 m/s²). The initial vertical velocity is set by the launch angle:
vy = v0y = v0 sin θ
As the projectile rises, vertical velocity decreases until it reaches zero at the peak of the trajectory. After that, the projectile falls, and vertical velocity increases in the downward direction.
Horizontal Motion | Vertical Motion | |
|---|---|---|
Acceleration | None (constant velocity) | Constant, g ≈ 9.81 m/s² downward |
Initial velocity | v0x = v0 cos θ | v0y = v0 sin θ |
At the peak | Unchanged | Zero |
Range and Time of Flight
The range is the horizontal distance a projectile travels before hitting the ground, depending on initial velocity and launch angle:
R = (v0² sin 2θ) / g
Because projectile motion is symmetric, the time to rise to maximum height equals the time to fall back down — so total flight time is simply twice the time to reach the peak:
t = (2v0 sin θ) / g
Inclined Planes
Inclined planes are a classic two-dimensional motion scenario that combines vectors, gravity, and Newton's laws of motion. The key to analyzing forces on an incline is breaking them down into components parallel and perpendicular to the plane's surface, which makes it far easier to see how forces interact and affect motion.
On an incline, gravity no longer acts straight down relative to the surface — it must be split into two components:
Parallel to the incline (pulls the object down the slope): mg sin θ
Perpendicular to the incline (presses the object into the surface): mg cos θ
where m is the object's mass, g is the acceleration due to gravity, and θ is the incline's angle from horizontal.
Uniform Circular Motion
Uniform circular motion describes an object traveling at constant speed along a circular path. Even though its speed doesn't change, its direction is continuously changing — which means the object is always accelerating. This acceleration, called centripetal acceleration, is directed toward the center of the circle, and it's what keeps the object moving in a circular path instead of a straight line.
Centripetal Force and Acceleration
Since velocity is a vector with both magnitude and direction, its constant change in direction requires a force pointed toward the center of the circle — the centripetal force. It's responsible for centripetal acceleration and is calculated as:
F = mv² / r
where m is the object's mass, v is its speed, and r is the radius of the circular path.
Why Centrifugal Force Isn't Real
As an object moves in a circular path, it might feel like there's a force pushing it outward — often called centrifugal force. But centrifugal force isn't a real force; it's a fictitious force that arises from the object's inertia — its tendency to keep moving in a straight line. From inside a rotating frame of reference, this feels like an outward push, but the only real force acting on the object is the centripetal force, pulling it inward.
Example: a car making a turn relies on friction between the tires and road to supply the necessary centripetal force. If the car moves too fast, or the curve is too sharp, the required centripetal force can exceed the available friction, causing the car to skid outward. This skidding is often mistakenly attributed to "centrifugal force," but it's actually the result of insufficient centripetal force to maintain the circular path.
Common MCAT Mistakes
Using the wrong kinematic equation for the wrong unknown. Each of the four equations omits a different variable — picking one that still requires an unknown quantity leads to unsolvable setups. Always check which variable a scenario doesn't give you, then pick the equation missing exactly that variable.
Treating horizontal and vertical motion as connected in projectile motion. They're independent — horizontal velocity never changes during flight (assuming no air resistance), and only vertical velocity is affected by gravity. Mixing g into a horizontal-velocity calculation is a common error.
Forgetting that vertical velocity is zero at the peak of a trajectory, not the total speed. At the top of a projectile's arc, only the vertical component is zero — the horizontal component (v0 cos θ) is still moving the object forward the entire time.
Believing centrifugal force is a real, outward-pushing force. The only real force in circular motion is centripetal, directed inward. What feels like an outward push is inertia — the object's tendency to keep moving in a straight line — resisted by the centripetal force pulling it into a curve.
MCAT-Style Concept Check
Question: A ball is thrown horizontally off a cliff at the same instant a second, identical ball is dropped straight down from the same height. Ignoring air resistance, which ball reaches the ground first?
A) The ball thrown horizontally, because its higher speed increases its downward acceleration.
B) The dropped ball, because it has no horizontal motion to slow its descent.
C) Both balls land at the same time, because horizontal and vertical motion are independent and both start with zero vertical velocity.
D) The dropped ball, because gravity acts on it alone, while the thrown ball must share gravity's pull between two directions of motion.
Answer: C
Explanation: Horizontal and vertical motion are independent in projectile motion. The thrown ball's horizontal velocity (v0 cos θ) has no effect on its vertical motion, which — like the dropped ball — starts at zero vertical velocity and accelerates downward at g ≈ 9.81 m/s². Since both balls share the same initial vertical velocity (zero) and the same vertical acceleration (g), they fall the same vertical distance in the same amount of time, landing simultaneously. Options A, B, and D all incorrectly assume horizontal motion or speed affects the rate of vertical fall — gravity acts equally on both balls regardless of their horizontal motion.
FAQ
How do I know which kinematic equation to use?
Identify which variable the problem doesn't give you and doesn't ask for — each of the four kinematic equations omits exactly one of the five variables (v₀, v, a, t, Δx). Pick the equation that leaves out the one you don't have.
Why do horizontal and vertical velocity behave differently in projectile motion?
Because they're governed by different forces. No horizontal force acts during flight (assuming negligible air resistance), so horizontal velocity stays constant. Gravity acts only vertically, so vertical velocity continuously changes — decreasing on the way up, reaching zero at the peak, then increasing on the way down.
What's the difference between the parallel and perpendicular components of gravity on an incline?
The parallel component (mg sin θ) acts along the slope's surface and is what accelerates the object down the ramp. The perpendicular component (mg cos θ) presses the object into the surface and is balanced by the normal force, keeping the object on the incline rather than pushing through it.
Is centrifugal force ever a real force?
No — in any inertial (non-rotating) reference frame, centrifugal force doesn't exist. It only "appears" as a fictitious force from inside a rotating frame of reference, as a way to explain the outward sensation caused by inertia. The only real force in uniform circular motion is centripetal force, directed toward the center.
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