AP Physics 1 solving kinematics and dynamics problems: a complete skills guide to vectors, motion graphs, kinematic equations, free-body diagrams and Newton's laws

Why this is the spine of AP Physics 1

Units 1 and 2 supply the problem-solving engine that every later unit borrows. Energy, momentum and rotation all begin with the same moves: choose a system, draw a diagram, set a sign convention, and reason from forces to motion. This guide ties together the matching dot-point pages, each of which has its own practice questions: scalars and vectors, displacement, velocity and acceleration, representing motion, vectors and motion in two dimensions, forces and free-body diagrams, Newton's second law, and circular motion.

Vectors and sign conventions

Every quantity is a scalar (magnitude only: distance, speed, time, mass) or a vector (magnitude and direction: displacement, velocity, acceleration, force). In one dimension, a vector's direction is carried by a sign: choose a positive direction and stick to it. In two dimensions, resolve each vector into perpendicular components with $V_x = V\cos\theta$ and $V_y = V\sin\theta$ (when $\theta$ is measured from the horizontal), work on each axis separately, then recombine with $V = \sqrt{V_x^2 + V_y^2}$. Stating your positive direction before substituting numbers prevents most sign errors.

The kinematic equations

For one-dimensional motion with constant acceleration, three equations link displacement, the two velocities, acceleration and time:

Each omits one variable (displacement, final velocity, time, respectively). List your knowns and the unknown, pick the equation with exactly those, and substitute. Split any motion whose acceleration changes (speed up, then cruise, then brake) into separate constant-acceleration segments.

Reading motion graphs

Graphs are tested heavily, and the rules are fixed:

• On a position-time graph, the slope is velocity; curvature means acceleration.
• On a velocity-time graph, the slope is acceleration and the area under the line is displacement.
• On an acceleration-time graph, the area under the line is the change in velocity.

Area below the time axis is negative. Always check which quantity is on the vertical axis before reading a slope or an area.

Projectile motion

A projectile moves under gravity alone after launch. Its horizontal and vertical motions are independent, linked only by the shared time of flight:

• Horizontal: no force, so $a_x = 0$ and $v_x$ is constant; $x = v_x t$.
• Vertical: constant acceleration $g$ downward, so the vertical motion is the free-fall problem.

Resolve the launch velocity into $v_{0x} = v_0\cos\theta$ and $v_{0y} = v_0\sin\theta$, find the time from the vertical axis, then use it horizontally. A ball thrown horizontally and one dropped from the same height land together, because their vertical motions are identical.

Free-body diagrams and Newton's laws

A free-body diagram shows only the forces acting on the chosen object, as labelled arrows from a point: weight ($mg$, down), normal force (perpendicular to the surface), friction (along it, opposing sliding), tension (along ropes), applied and spring forces. From the diagram, find the net force and apply Newton's laws:

• First law: zero net force means constant velocity (including rest). Equilibrium problems set $\sum F_x = 0$ and $\sum F_y = 0$.
• Second law: $F_{net} = ma$, applied axis by axis ($F_{net,x} = m a_x$, $F_{net,y} = m a_y$).
• Third law: forces come in equal-and-opposite pairs on different objects, so a pair never cancels and never shares a diagram.

For a connected system, treat the whole as one mass for the common acceleration, then a single block for the internal tension.

Friction, gravity, springs and circular motion

These are the specific forces that fill in a free-body diagram:

• Friction: kinetic friction is fixed, $f_k = \mu_k N$; static friction adjusts up to a maximum $f_{s,max} = \mu_s N$. Compare an applied force with $\mu_s N$ to decide whether motion starts.
• Gravity: $F = \dfrac{Gm_1 m_2}{r^2}$ (inverse-square); near a planet the field strength is $g = \dfrac{GM}{r^2}$ and weight is $W = mg$. Mass is fixed; weight varies with $g$.
• Springs: Hooke's law gives a restoring force of magnitude $F = kx$ toward equilibrium.
• Circular motion: uniform circular motion has centripetal acceleration $a_c = \dfrac{v^2}{r}$ toward the center, requiring a net inward force $F_c = \dfrac{mv^2}{r}$ supplied by a real force (tension, gravity, friction, normal). There is no outward force.

Check your knowledge

A mix of recall, calculation and application questions covering Units 1 and 2. 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 car accelerates uniformly from $10$ m/s to $30$ m/s in $5.0$ s. Calculate its acceleration. (2 marks)
3. On a velocity-time graph, state what the slope and the area under the line represent. (2 marks)
4. A ball is thrown horizontally at $6.0$ m/s from a $1.8$ m high table ($g = 9.8$ m/s squared). Calculate the time to land. (2 marks)
5. State what a free-body diagram should never include. (1 mark)
6. A $3.0$ kg object has a net force of $12$ N. Calculate its acceleration. (2 marks)
7. Explain why action-reaction forces do not cancel. (2 marks)
8. A $6.0$ kg box on a level floor has $\mu_k = 0.25$. Calculate the kinetic friction force ($g = 9.8$ m/s squared). (2 marks)
9. State what happens to the gravitational force between two masses if the distance between them is tripled. (1 mark)
10. A $0.40$ kg ball moves in a circle of radius $1.5$ m at $3.0$ m/s. Calculate the centripetal force. (2 marks)