Regents Physics kinematics: a complete skills guide to vectors, motion graphs, the kinematic equations, free fall and projectile motion

Why kinematics is the foundation of Regents Physics

Kinematics is the language the rest of the course is written in. Forces are studied to explain accelerations; energy and momentum track how motion changes; circular motion and gravitation are kinematics with a particular acceleration. Master the moves here, choosing a direction, picking the right equation from the Reference Tables, and reading motion graphs, and the mechanics half of the exam becomes routine. This guide ties together the dot-point pages, each with its own practice: vectors and scalars, displacement, velocity and acceleration, graphs of motion, the kinematic equations, free fall and projectile motion.

Scalars, vectors and sign conventions

Every quantity is a scalar (magnitude only: distance, speed, mass, time, energy) or a vector (magnitude and direction: displacement, velocity, acceleration, force, momentum). In one dimension a vector's direction is carried by a sign: choose a positive direction and apply it consistently, so a negative answer simply means the other way. In two dimensions, resolve a vector into perpendicular components with $A_x = A\cos\theta$ and $A_y = A\sin\theta$ (angle from the horizontal), work on each axis, then recombine with $A = \sqrt{A_x^2 + A_y^2}$. Perpendicular vectors add by the Pythagorean theorem, so $6$ east and $8$ north give a resultant of $10$, not $14$. None of these vector formulas is printed on the Reference Tables; you supply them.

The kinematic equations and how to choose one

For motion with constant acceleration, the Reference Tables print:

together with $\bar{v} = \dfrac{d}{t}$, from which $d = \tfrac{1}{2}(v_i + v_f)t$ follows. Each of the first three omits one quantity: the first has no displacement, the second no final velocity, the third no time. The selection rule is therefore simple: list the three quantities you know and the one you want, and choose the equation that omits the fifth. This is faster and safer than guessing. The equations apply only while acceleration is constant, so split multi-stage motion (accelerate, cruise, brake) into separate stages.

Reading motion graphs

Graphs appear in Parts A, B-1 and B-2. The rules are fixed:

• Position-time: the slope is the velocity; a curve means changing velocity (acceleration).
• Velocity-time: the slope is the acceleration and the area under the line is the displacement.
• Acceleration-time: the area under the line is the change in velocity.

A horizontal line means the plotted quantity is constant; area below the time axis is negative. In a data-plotting task, draw a best-fit line through the trend of the points (not dot to dot), and calculate any slope from two points on the line, interpreting it as the physical rate.

Free fall

Free fall is motion under gravity alone, with air resistance neglected. Every object then has the same constant acceleration, $g = 9.81$ m/s squared downward, regardless of mass, so the kinematic equations apply with $a = g$. A dropped object starts from $v_i = 0$; an object thrown up decelerates, stops momentarily at the top (where $v = 0$ but the acceleration is still $g$ downward), and falls back symmetrically. The value of $g$ is on the Reference Tables, and the weight relation $F_g = mg$ links free fall to forces.

Projectile motion

A projectile is acted on only by gravity after launch, so the motion splits into two independent parts joined by a shared time:

• Horizontal: no force, constant velocity, $d_x = v_x t$.
• Vertical: free fall, $g$ downward; launched horizontally, $d_y = \tfrac{1}{2}gt^2$.

Solve the vertical motion first to find the time of flight (set by the drop height), then use that time in the horizontal equation for the range. The landing speed combines the horizontal and vertical velocities as perpendicular components, $v = \sqrt{v_x^2 + v_{fy}^2}$.

Check your knowledge

A mix of recall, calculation and graph questions covering the kinematics module. Attempt them under timed conditions, then check against the solutions.

1. State the difference between a scalar and a vector, with one example of each. (2 marks)
2. A walker goes $9.0$ m east then $12$ m north. Calculate the magnitude of the resultant displacement. (2 marks)
3. On a velocity-time graph, state what the slope and the area under the line represent. (2 marks)
4. A car accelerates from $5.0$ m/s to $25$ m/s in $4.0$ s. Calculate its acceleration. (2 marks)
5. State which kinematic equation omits the final velocity. (1 mark)
6. An object is dropped from rest. Calculate the distance it falls in $2.0$ s ($g = 9.81$ m/s squared). (2 marks)
7. State the acceleration of a freely falling object at the highest point of its rise. (1 mark)
8. A ball is launched horizontally at $10.$ m/s and is in the air for $1.5$ s. Calculate the horizontal range. (2 marks)
9. Explain why a projectile's horizontal velocity stays constant. (2 marks)
10. A car travels at constant $15$ m/s for $8.0$ s. Calculate its displacement using a graph idea. (2 marks)