Not all motion takes place along a straight line. A ball thrown at an angle to the horizontal moves in a curved path, an airplane changes both direction and speed, and a satellite orbits the Earth along a curved trajectory. To describe such motion we need vectors and the mathematics of two dimensions. This chapter extends the ideas of position, velocity, and acceleration to motion in a plane.
We begin with the basic operations of vector algebra - addition, subtraction, and resolution of vectors into components. We then study projectile motion, in which a body moves under the influence of gravity alone, and uniform circular motion, in which a body moves along a circle with constant speed. We close with the concept of relative velocity in two dimensions.
Vectors are essential because quantities like displacement, velocity, and acceleration have both magnitude and direction. By resolving vectors into mutually perpendicular components, the complicated two-dimensional motion of a projectile is split into two independent one-dimensional motions, one along the horizontal and one along the vertical.
A scalar quantity has only magnitude and no direction. Mass, time, temperature, distance, speed, and energy are scalars. A vector quantity has both magnitude and direction. Displacement, velocity, acceleration, and force are vectors. Vectors are represented by arrows: the length of the arrow gives the magnitude and the arrowhead gives the direction.
A vector A in a plane can be resolved into two perpendicular components along the x and y axes. If A makes an angle theta with the positive x-axis, then:
Ax = A cos theta
Ay = A sin theta
The magnitude of A is given by:
A = sqrt(Ax^2 + Ay^2)
and the direction angle is found from tan theta = Ay / Ax.
Vectors are added by the triangle law or the parallelogram law. By the parallelogram law, if two vectors A and B are represented by adjacent sides of a parallelogram, their resultant R is given by the diagonal through the common point. The magnitude of the resultant is:
R = sqrt(A^2 + B^2 + 2AB cos theta)
where theta is the angle between A and B. The direction of R makes an angle alpha with A, given by:
tan alpha = (B sin theta) / (A + B cos theta)
If A and B are perpendicular, then R = sqrt(A^2 + B^2). The resultant is maximum (A + B) when the vectors are parallel and minimum (|A - B|) when they are antiparallel. To subtract a vector B from A, we add the negative of B, that is A - B = A + (-B).
For addition using components, if R = A + B, then Rx = Ax + Bx and Ry = Ay + By.
The position of a particle in a plane is described by a position vector r = x i + y j. The velocity is the time rate of change of the position vector:
v = dr/dt = dx/dt i + dy/dt j
and acceleration is:
a = dv/dt = d^2r/dt^2
Relative velocity in two dimensions follows the same vector rule as in one dimension. The velocity of A relative to B is:
v_AB = v_A - v_B
where v_A and v_B are the velocities of A and B with respect to the ground. For example, if two cars move with velocities v1 and v2 at an angle to each other, an observer in car 2 sees car 1 moving with velocity v1 - v2.
Projectile motion is the motion of a body projected into the air, moving freely under gravity after being given an initial velocity. A cricket ball hit into the air and a shell fired from a cannon are projectiles. The horizontal and vertical motions are independent.
If a projectile is launched with velocity u at an angle theta to the horizontal, its components are ux = u cos theta and uy = u sin theta. The horizontal motion is uniform (acceleration zero) while the vertical motion has acceleration -g. The horizontal range is:
R = (u^2 sin 2theta) / g
The maximum height reached is:
H = (u^2 sin^2 theta) / (2g)
and the total time of flight is:
T = (2u sin theta) / g
The trajectory is a parabola. The range is maximum (equal to u^2/g) when theta = 45 degrees. The range is the same for angles theta and (90 - theta).
When a particle moves along a circular path with constant speed, its motion is called uniform circular motion. Although the speed is constant, the direction of velocity changes continuously, so the particle has acceleration. This acceleration is directed toward the centre of the circle and is called centripetal acceleration.
The centripetal acceleration is:
a_c = v^2 / r
where v is the speed and r is the radius of the circle. The angular velocity omega is the rate of change of angular displacement:
omega = 2 pi / T
where T is the time period. The relation between linear speed and angular velocity is:
v = omega * r
If the speed is not constant, the particle also has a tangential acceleration, and the total acceleration is the vector sum of the tangential and centripetal accelerations.
| Quantity | Scalar or Vector | Example |
|---|---|---|
| Speed | Scalar | 40 km/h |
| Velocity | Vector | 40 km/h north |
| Distance | Scalar | 5 m |
| Displacement | Vector | 5 m east |
| Mass | Scalar | 2 kg |
| Acceleration | Vector | 3 m/s^2 downward |
| Projectile Quantity | Formula | At theta = 45 degrees |
|---|---|---|
| Time of flight | T = 2u sin theta / g | u sqrt(2)/g |
| Maximum height | H = u^2 sin^2 theta / 2g | u^2 / 4g |
| Horizontal range | R = u^2 sin 2theta / g | u^2 / g |
| Trajectory | Parabola | Parabola |
In this chapter we extended kinematics to two dimensions using vectors. We learned to resolve vectors into components, to add them by the parallelogram law, and to compute relative velocity in a plane. Projectile motion showed us how the horizontal and vertical motions combine to produce a parabolic trajectory, with formulas for range, height, and time of flight. Uniform circular motion revealed that a body moving with constant speed can still be accelerating, with centripetal acceleration a_c = v^2/r directed toward the centre. These vector methods prepare us for the study of dynamics, where we relate the forces that cause such motions, in the laws of motion.