Motion in a Plane

The One Thing

Motion in a plane is analysed by representing physical quantities as vectors and resolving them into perpendicular components. This allows projectile motion and circular motion to be treated using separate horizontal and vertical equations while accounting for changes in both magnitude and direction.

Reactions, Processes and Experiments

Key Terms

• Scalar: A physical quantity described only by magnitude, such as distance, speed, and time.
• Vector: A physical quantity described by both magnitude and direction, such as displacement, velocity, and acceleration.
• Position Vector: A vector drawn from the origin to the position of a particle, usually written as r = xi + yj.
• Displacement Vector: The change in position of an object, given by the final position vector minus the initial position vector.
• Unit Vector: A vector having magnitude one and pointing in a specified direction; i, j, and k represent unit vectors along the x-, y-, and z-axes.
• Vector Addition: The process of finding the resultant vector using the triangle law or parallelogram law.
• Vector Components: The parts of a vector along chosen coordinate axes. A vector A making angle theta with the x-axis has components Ax = A cos theta and Ay = A sin theta.
• Projectile: An object projected into the air that moves under the influence of gravity alone, neglecting air resistance.
• Projectile Motion: Two-dimensional motion in which horizontal motion is uniform and vertical motion has constant downward acceleration g.
• Trajectory: The path followed by a moving object. The trajectory of an ideal projectile is a parabola.
• Range: The horizontal distance travelled by a projectile before it returns to the same level from which it was projected.
• Uniform Circular Motion: Motion along a circular path with constant speed, while the velocity changes continuously because its direction changes.
• Centripetal Acceleration: The inward acceleration required for circular motion, directed toward the centre of the circle.
• Angular Velocity: The rate of change of angular displacement, represented by omega and related to linear speed by v = r omega.

Easily Confused

• Scalar and vector: A scalar has magnitude only, whereas a vector has both magnitude and direction.
• Speed and velocity: Speed is a scalar, whereas velocity is a vector; in uniform circular motion, speed remains constant but velocity changes because its direction changes.
• Horizontal and vertical projectile motion: Horizontal motion is uniform, while vertical motion has constant downward acceleration g.
• Centripetal acceleration and centripetal force: Centripetal acceleration is the inward acceleration, whereas centripetal force is the net inward force producing that acceleration.
• Centripetal force and a separate force type: Centripetal force is not a new type of force; it is provided by forces such as tension, friction, gravity, or the normal reaction.
• Time of flight and time period: Time of flight refers to a projectile’s motion, whereas the time period is the time for one complete revolution in circular motion.
• Range and maximum height: Range is the horizontal distance travelled, whereas maximum height is the greatest vertical displacement above the launch level.
• General projection and horizontal projection: General projectile motion begins with components u cos theta and u sin theta, whereas horizontal projection has an initial vertical component of zero and is specified by height h and horizontal speed u.

What Gets Asked

• Resolve a vector into perpendicular components using Ax = A cos theta and Ay = A sin theta. Marks are lost by interchanging the sine and cosine components or failing to identify the angle’s reference axis.
• Calculate the resultant of two vectors using R = sqrt(A^2 + B^2 + 2AB cos theta), or distinguish the scalar product from the vector product. Marks are lost by treating A dot B as a vector or omitting the perpendicular direction n from A cross B.
• Derive or apply projectile equations for displacement, velocity, trajectory, time of flight, maximum height and range. Marks are lost by using same-level formulas when the launch and landing levels are not the same, or by confusing horizontal and vertical components.
• Determine the angle for maximum projectile range. The required condition is 45 degrees only when air resistance is neglected and the launch and landing levels are the same.
• Solve horizontal-projection problems using t = sqrt(2h/g) and x = u sqrt(2h/g). Marks are lost by applying the angled-projectile time-of-flight formula instead.
• Explain uniform circular motion and calculate a_c, F_c, T or f. Marks are lost by claiming that acceleration is zero because speed is constant, or by treating centripetal force as a separate fundamental type of force.