Motion in Two and Three Dimensions Explained | Chapter 3 of University Physics

Chapter 3 of University Physics advances kinematics into the realm of two- and three-dimensional motion. Using vectors to describe position, velocity, and acceleration, this chapter lays the mathematical foundation for understanding parabolic trajectories, circular motion, and the critical role of frames of reference in real-world physics.

Position, Displacement, and Velocity Vectors

To describe motion in space, we use the position vector r⃗, which specifies an object’s location relative to an origin. Displacement (Δr⃗) is the change in position, and is itself a vector with both magnitude and direction. Average velocity is displacement divided by time, while instantaneous velocity is the derivative of position with respect to time—always tangent to the path of motion. The speed is simply the magnitude of velocity, regardless of direction.

Acceleration in Two and Three Dimensions

Acceleration vectors are defined as the derivative of velocity and can be decomposed into Cartesian components. Importantly, even if the speed remains constant, acceleration is nonzero whenever the direction of velocity changes—as with objects following curved paths. Acceleration can be split into components parallel to velocity (changing speed) and perpendicular to velocity (changing direction).

Projectile Motion: Parabolic Trajectories

A major highlight is projectile motion, where an object moves in a parabola under constant gravitational acceleration. Here, horizontal and vertical motions are independent:

• Horizontal: ax = 0, velocity is constant.
• Vertical: ay = –g, constant downward acceleration due to gravity.

Kinematic equations allow you to solve for position and velocity over time. The range and maximum height can be calculated using trigonometric functions of the launch angle, with optimal range at 45° (when launch and landing heights are equal).

Circular Motion: Uniform and Nonuniform

Uniform circular motion involves movement at constant speed along a circle, but the direction changes continuously, producing centripetal (radial) acceleration toward the center:

• arad = v²/R
• arad = 4π²R/T²

For nonuniform circular motion, there is also a tangential acceleration component that changes the speed in addition to the radial component that changes direction.

Frames of Reference and Relative Velocity

Motion is always described relative to a frame of reference. The concept of relative velocity explains how an observer’s motion can change the measured velocity of another object. Vector equations help relate velocities across different frames in both one and two dimensions—key for solving problems involving moving platforms, vehicles, or fluids.

Conclusion

Chapter 3 provides the analytical and graphical tools to describe motion in multiple dimensions, whether for projectiles, circular paths, or moving observers. Mastering these concepts prepares you for advanced topics in mechanics, dynamics, and real-world problem solving in engineering and science.

To see vector equations and step-by-step worked problems, watch the full video summary here. Keep learning with Last Minute Lecture by exploring more chapter guides and subscribing for timely academic support.

If you found this breakdown helpful, be sure to subscribe to Last Minute Lecture for more chapter-by-chapter textbook summaries and academic study guides.