Rectilinear Motion

The kind of motion that was discussed above is called rectilinear motion, which refers to the motion of an object in a straight line. Such motion can be depicted as a point which moves forwards and/or backwards on a number line.

General Motion Equations

If s(t) represents the position of the object on the number line at time t, then v(t), the (instantaneous) velocity, is equal to s'(t), and a(t), the (instantaneous) acceleration, is equal to v'(t), which is s''(t).

Thus, velocity is the rate of change of position, and acceleration is the rate of change of velocity.

Example:

If s(t) = t2 - 5t, what is the position, velocity and acceleration at t = 2? Assume s is in feet and t is in seconds, and interpret these results.

s(t) = t2 - 5t + 3
v(t) = s'(t) = 2t - 5
a(t) = v'(t) = 2

s(2) = 2
v(2) = - 1
a(2) = 2

So, at t = 2, the object is located at +2 feet from the start. The velocity is -1 foot per second. The negative sign indicates that it is headed in the negative direction, and it is moving backwards at a rate of one foot per second. The acceleration is 2, which means that at that instant, its velocity is increasing by a rate of 2 feet per second each second.

Vector and Scalar Quantities

Position, velocity, and acceleration are all vector quantities because they contain both a direction and a magnitude. For example, if the velocity of an object is -3 feet per second, then that object is moving backwards (direction) at a rate of 3 feet per second (magnitude). Similarly, if an object has a position of -3 feet, then is located 3 feet from the starting point (magnitude), but on the negative side (direction).

The vector quantities of position and velocity both have corresponding scalar quantities that only have a magnitude. The scalar analog of position is distance. Although the position of an object with respect to the start line may be -3 feet, its distance from that start line is simply 3 feet, because distance is always a positive quantity. Thus, distance is the absolute value of position.

Similarly, the scalar analog of velocity is speed. Whether an object's velocity is -5 feet per second, or +5 feet per second, its speed is still simply 5 feet per second, because speed is always a positive quantity that contains no information about direction. Thus, speed is the absolute value of velocity.