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doPhysics · Kinematics

Motion in Two Dimensions

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Why this chapter
Why this chapter exists

A cricket ball, a stream of water from a hose, a bike turning at a corner: none of these moves along one straight line. The last chapter gave you position, velocity and acceleration on a line. This chapter gives the same three quantities a direction in a plane, and then shows that one two-dimensional motion is only two one-dimensional motions happening at the same time. That single idea makes projectile motion easy.

You will learn
By the end of this chapter you will
  1. Write position, displacement, velocity and acceleration in a plane as vectors with components.
  2. Say why the velocity is always tangent to the path, and what an acceleration does to the speed and to the direction.
  3. Treat a projectile as two independent motions, horizontal at constant velocity and vertical under gravity.
  4. Derive and use the time of flight, the maximum height, the range and the equation of the trajectory.
  5. Use the equations of motion in vector form, and solve a launch from a height aimed up, level or down.
  6. Read the launch speed, the angle and the range from a given equation of the path, and say when the parabola fails.
Before you begin
Before you begin

You should know the whole of Motion in One Dimension, especially the three equations of motion and free fall with one sign convention. You should be able to add two vectors, split a vector into components along two perpendicular axes, and use sine and cosine for a given angle.

Already know this?

Lessons

1
Position and Displacement in a Plane
Two numbers locate a body in a plane. The position vector, its components, and the displacement vector that joins two positions and ignores the path between them.
Position and displacement as vectors
12 min0%
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2
Velocity and Acceleration in a Plane
Average and instantaneous velocity as vectors, why the velocity always points along the tangent to the path, and how an acceleration can change the speed, the direction, or both, and the equations of motion written as vectors.
Velocity is tangent to the pathAcceleration changes speed or directionEquations of motion in vector form
19 min0%
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3
Two Motions at Once
The idea that makes projectiles simple. The horizontal and vertical motions of a body are independent, each with its own equations, joined only by the common time.
Perpendicular motions are independent
12 min0%
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4
Projectile Motion
A body thrown at an angle, treated as two motions. The components of the velocity, the time of flight, the maximum height and the range, each derived from the equations of motion.
Projectile as two motionsTime of flight height and range
16 min0%
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5
The Trajectory and Its Special Cases
The equation of the path, why it is a parabola, the angle that throws farthest, the two angles that share a range, and a projectile launched from a height, level or at an angle, aimed up, level or down.
The equation of the trajectoryRange and the angle of projectionProjection from a height
21 min0%
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6
The Trajectory as a Tool
The trajectory in factored form, its two roots and its symmetry, reading the launch speed, the angle and the range from a given equation of the path, and the limits of the parabola.
Factored form of the trajectoryReading a launch from the trajectoryLimits of the parabola
16 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

air drag
The push of the air on a moving body, directed opposite to its velocity and growing with speed. It shortens and lowers a real trajectory.
angle of projection
The angle between the initial velocity of a projectile and the horizontal.
component
The part of a vector along one axis. A vector in a plane has an \(x\) component and a \(y\) component.
displacement
The change in position, the vector from the initial position to the final position. It does not depend on the path.
factored form
The trajectory written as \(y = x\tan\theta\,(1 - x/R)\). It shows the two points where the path meets the ground, \(x = 0\) and \(x = R\).
independence of motions
The principle that the horizontal and vertical motions of a body affect each other in no way. Each is solved on its own with its own equations.
maximum height
The greatest height above the launch point reached by a projectile.
position
Where a body is, given by its coordinates measured from an origin. In a plane a position needs two numbers.
position vector
The vector from the origin to the position of a body, written \(\vec r = x\,\hat i + y\,\hat j\) in a plane.
projectile
A body thrown into the air and then moving under gravity alone. Air resistance is neglected.
range
The horizontal distance between the launch point and the landing point of a projectile on the same level.
root
A value of \(x\) at which an expression equals zero. The roots of the trajectory are the points where the path is at the launch level.
tangent
The straight line that touches a curve at one point and follows the direction of the curve there. Velocity is always along the tangent to the path.
time of flight
The total time a projectile spends in the air between launch and landing.
trajectory
The path traced by a moving body. For a projectile near the Earth the trajectory is a parabola.
unit vector
A vector of magnitude one used to show a direction. \(\hat i\) points along the \(x\) axis and \(\hat j\) along the \(y\) axis.