## Basic Concepts

A few basic concepts are necessary to understand Newton’s laws and their implications. Some of these concepts will likely be familiar to you already, but a quick review may be helpful nonetheless. Let’s get right into it!

A scalar quantity is simply an amount (or magnitude) of something, represented by a number with units. Speed, mass, and energy are examples of scalar quantities.

A vector quantity has both magnitude and direction, and is represented by an arrow. The length of the arrow represents the magnitude, and the direction the arrow is pointing represents the direction of the vector quantity. For example, if an object is moving 5 meters per second to the left, its velocity can be represented by an arrow 5 units long pointing to the left. Velocity, acceleration, momentum, and force are examples of vector quantities.

To add vector quantities together, you place the front tip of one vector at the tail end of the other, then draw a new vector from the tail of the first to the tip of the second. This new vector is the sum of the two original vectors. You can think of vector addition as analogous to walking: if you walk a certain distance in one direction, then walk a certain distance in another direction, your destination will be located at the sum of those two vectors.

If a 1-unit-long vector and a 2-unit-long vector both point to the right, their sum is a 3-unit-long vector pointing right. To visualize this, imagine walking 1 meter to the right, then walking another 2 meters in the same direction. You’ll end up 3 meters to the right of where you started.

If a 10-unit vector points right and a 15-unit vector points left, their sum is a 5-unit vector pointing left. Likewise, if you walk 10 meters to the right and then walk 15 meters in the opposite direction, you’ll end up 5 meters to the left of where you started.

If a 4-unit vector points east and 3-unit vector points north, their sum is a 5-unit vector pointing northeast (at an angle approximately 37 degrees from east), as shown in the following illustration: Analogously, if you walk 4 meters east and then walk 3 meters north, you’ll end up 5 meters northeast of where you began.

Vector addition is even easier when the vectors are placed in a coordinate system with their tails at the origin. By placing all of the vector tails at the origin, you can find the sum of two (or more) vectors simply by adding their coordinates for each axis.

For example, suppose two vectors both have their tails at the origin (0,0,0) in a three-dimensional coordinate space. The tip of the first vector is located at the point (x1,y1,z1) and the tip of the second is located at the point (x2,y2,z2). The sum of those two vectors is a vector with its tip located at the point (x1+x2, y1+y2, z1+z2).

If the original vectors have coordinates (1,3,5) and (2,4,6), for instance, their sum will have coordinates (3,7,11).

As mentioned above, velocity is a vector quantity; speed is a scalar quantity. Speed is simply the rate at which something is moving, without regard to its direction of motion. Velocity is the rate of motion in a specific direction.

Acceleration is the rate and direction of a change in velocity. An object can accelerate by speeding up, slowing down (this is usually called deceleration), or changing direction. Like velocity, acceleration is a vector quantity: it has both magnitude and direction. The magnitude of the acceleration is the rate at which the velocity is changing. The direction of the acceleration vector is the direction toward which the velocity vector is changing or turning:

• If an object is speeding up (without changing direction), the acceleration vector points in the same direction as the velocity vector.
• If an object is slowing down (without changing direction), the acceleration vector points in the opposite direction from the velocity vector.
• If an object is changing direction (without changing speed), the acceleration vector is perpendicular to the velocity vector and points in the direction toward which the velocity is turning.

Objects can accelerate (change velocity) without speeding up or slowing down. When an object’s acceleration is perpendicular to its velocity, the direction will change while the speed remains constant. For example, the moon is accelerating toward the earth as it orbits, though its speed remains approximately constant at about 1 km/s. The moon is accelerating toward us, but fortunately it isn’t getting any faster or closer! Its direction of motion is always tangential to its orbit, and perpendicular to its acceleration toward the earth.

The momentum of an object is its mass times its velocity. It too is a vector quantity. The total momentum of a group of objects is the sum of their individual momentum vectors. (See above for an explanation of how vectors are added together.)

Equations Standard units
speed =
 distance time
meters per second (m/s)
velocity =
 change in position time
meters per second (m/s)
acceleration =
 change in velocity time
meters per second, per second (m/s2)
momentum = mass × velocity kilogram meters per second (kg m/s)
The above equations are for linear velocity, acceleration, and momentum. Angular velocity is the rate and direction of an object’s spin, and angular acceleration is the rate of change in angular velocity. Angular momentum is equal to angular velocity times rotational inertia, which depends on how the mass of an object is distributed. (When the mass is spread out from the axis of rotation, the rotational inertia is greater than when the mass is close to the axis of rotation.)

Notice that the equation for acceleration implies that change in velocity = acceleration × time So, if an object begins at rest (velocity zero) and accelerates at a constant rate a for t seconds, its final velocity is: a × t.

When an object begins at rest and accelerates at a constant rate, the distance it travels can be calculated by the following equation: distance = ½ × acceleration × time2