If you know an object is moving with constant velocity v, can you predict the position of the object at any later time t? How? How do the equations we have developed so far lead to a model of constant-speed motion?

The position model

The answer is to solve the velocity equation for the displacement Δx, then break Δx into the initial position x_{i} and final position x_{f}. The result is equation (3.5), which is really just a restatement of equation (3.3).

where x_{0} is the initial position at time t = 0.

(3.5)

$${x}_{f}={x}_{i}+vt$$

x_{f}

=

position at time t (m)

x_{i}

=

initial position at time t = 0 (m)

v

=

velocity (m/s)

t

=

time (s)

Position constant velocity

The position, time, and velocity in equation (3.5) are instantaneous values. This equation allows you to calculate the instantaneous position at any time when given the initial position and (constant) velocity.

An example problem

A train leaves a station located 300 km from the start of a track. How far is the train from the start of the track if it travels for 8 hr at 100 km/hr?

Asked:

final position x_{f} of the train

Given:

v = 100 km/hr; x_{i} = 300 km; t = 8 hr

Relationships:

x_{f} = x_{i} + vt

Solution:

The units are km/hr and hours, which are consistent:

x = 300 km + (100 km/hr)(8 hr) = 1,100 km

Answer:

1,100 km

What if the velocity changes?

You might think the limitation to constant velocity makes equation (3.5) useless in the real world. Nonetheless, complex motion can be accurately modeled by breaking up the time into intervals. The velocity is recalculated for each interval and is only considered constant during that interval. The final position for one interval becomes the initial position for the next interval. Many computer models use this approach, as do robots and self-navigating vehicles.