One-Dimensional Kinematics

One-dimensional kinematics is the study of motion in one dimension.

Kinematics vs Dynamics

Kinematics studies how objects move, considering the various trajectories and velocities.

On the other hand, dynamics studies the causes of motion, the forces that cause objects to move.

Position

Position is the location of an object in space. It is a vector quantity, meaning it has both magnitude and direction.

Usually, position is denoted as a function of time , where is time. In one-dimensional kinematics, the position can be written as . Here, is the unit vector in the direction of motion.

Displacement

Displacement is the change in position of an object, or . It is also a vector quantity. Displacement can be expressed as follows:

In one-dimensional kinematics, the displacement can be written as:

Velocity

Velocity is the rate of change of position with respect to time. It is a vector quantity. The SI unit of velocity is meters per second ().

The average velocity, which is the average rate of change of position, is given by:

In one-dimensional kinematics, the average velocity can be written as:

The instantaneous velocity is the velocity at a specific point in time. It is the limit of the average velocity as the time interval approaches zero:

The vector form of velocity is:

And in one-dimensional kinematics:

Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity. The SI unit of acceleration is meters per second squared ().

The average acceleration, which is the average rate of change of velocity, is given by:

In one-dimensional kinematics, the average acceleration can be written as:

Just like velocity, the instantaneous acceleration is the acceleration at a specific point in time. It is the limit of the average acceleration as the time interval approaches zero:

The vector form of acceleration is:

And in one-dimensional kinematics:

Constant Acceleration

If acceleration is constant, problems are very easy to solve.

Let where is a constant.

For a constant slope of , the velocity-time graph is a straight line. It can move anywhere vertically. Let the initial velocity be . Sometimes is written as .

Shown above is the velocity-time graph for a constant acceleration. The displacement of the object is the area under the graph, which can be divided into two parts: and . is the initial position of the object. Changing does not affect the velocity-time graph.

In vector form in one-dimensional kinematics, the displacement can be written as:

Example: Vehicles

Imagine a train traveling at a constant velocity of . Then, imagine a road parallel to the train tracks, with a car stopping at a traffic light.

The moment the traffic light turns green, the train passes the car, and the car starts accelerating at . At what time does the car catch up to the train?

To solve this problem, first evaluate the position functions for both the train and the car.

For the train, the position function is:

For the car, the position function is:

When the car catches up to the train, that means the positions are equal, so the lines on the graph intersect.

The car catches up to the train at . (Technically the algebra neglected as a solution, which is the initial position of the car.)

To find out how far the car has traveled, substitute into the car's position function:

The car has traveled when it catches up to the train.

Non-Constant Acceleration

For non-constant acceleration, the velocity-time graph is not a straight line.

To get the velocity, we need to make use of calculus.

This is an antiderivative, which is the reverse of differentiation. For example, if where is a constant, then:

Consider an acceleration-time graph. The area under the graph is the velocity. The area is thought of as a sum of rectangles, each with a width of and a height of . To find the area, notice that a small change in time incurs a small change in area, which is a rectangle and as such can be calculated as .

And therefore,

When , the area from to is 0, so . Hence,

This is the fundamental theorem of calculus.

Example: Bicycle and Car

A car travels at a constant velocity of .

At , the car starts braking, and at , the car stops.

The acceleration of the car can be expressed in this piecewise function:

Where .

Find the position function of the car.

A bicycle starts 15 meters behind the car and travels at a constant velocity of . When the car stops, the bicycle passes the car. Find the velocity of the bicycle.

(Adapted from MIT OCW 8.01)

To find the position function of the car, we need to integrate the acceleration function.

For :

For , it's a bit more complicated.

Beforehand, notice that because the car only starts braking at , so at that instant it doesn't change velocity yet. Firstly, the velocity can be found by constructing an equation through the fundamental theorem of calculus (Note the in the integral is a dummy variable):

Then, the position function can be found by integrating the velocity function:

Let , then :

Therefore, the position function of the car is:

To find the velocity of the bicycle, we need to find the time at which the car stops.

At , . At , , and .

Using the previous velocity function:

Note that this implies:

Therefore, the position at which the car stops is:

Since , we can construct an equation for the bicycle's position:

The velocity of the bicycle is .

Summary

• Position is the location of an object in space.
• Displacement is the change in position of an object.
• Velocity is the rate of change of position with respect to time.
• Acceleration is the rate of change of velocity with respect to time.
• For constant acceleration, the velocity-time graph is a straight line, and the position function is .
• For non-constant acceleration, the velocity-time graph is not a straight line.
• In differential order: position velocity acceleration; .
• In integral order: acceleration velocity position; .