One Dimensional Motion

Student Expectation

The student is expected to describe and analyze motion in one dimension using equations with the concepts of distance, displacement, speed, average velocity, instantaneous velocity, and acceleration.

Key Concepts

    • The position and motion of an object are described using vectors. A vector is a quantity that has both a magnitude and a direction. Vectors can be represented graphically using arrows, as well as verbally or mathematically.

    • The displacement of an object is the change in position in space compared to a chosen point of origin. The displacement may be different from the total distance the object has moved.

    • Both the displacement of an object and the distance an object has moved can be measured directly or calculated using the elapsed time and motion equations.

    • The motion of an object can be described either by its average motion (average velocity) or by its instantaneous motion (instantaneous velocity). The average velocity of an object is its change in position divided by the time interval of the motion. The instantaneous velocity of an object is the velocity at a single point in time and space. The instantaneous velocity is shown graphically as the slope of the tangent line on a position vs. time graph.

    • Speed is a scalar quantity; it has a magnitude, but not a direction. The average speed of a motion is the total distance travelled divided by the time interval of the motion.

    • If an object is speeding up, slowing down, or changing direction, then its motion (i.e. velocity) is not constant in time. In one-dimensional motion, acceleration indicates how fast an object’s velocity is changing, either speeding up (accelerating), slowing down (decelerating), or changing direction.

ONE DIMENSIONAL MOTION

Motion Described by Scalar or Vector Quantities

If motion happens along only one axis, either vertical or horizontal, it is called one-dimensional motion. It can be described using both scalar and vector quantities. Scalar quantities are described by a magnitude alone such as speed (calculated by dividing distance traveled by time). Vector quantities require both a magnitude and direction. The motion of an object can be illustrated in a graph or can be expressed in a table:

The position and motion of an object are related to a point of origin by an arrow showing direction and magnitude. This is called a vector quantity and can be used to interpret different kinds of motion graphically. In the example from the beginning table, the speed at 1 is 62km/h. This is a scalar quantity, since it only has magnitude. If it also indicates the direction, such as a car driving east on Westheimer Road toward Loop 610, it becomes a vector quantity. The vector quantity has both a magnitude and direction, while the scalar quantity shows magnitude only.

How far someone traveled can be described by distance and displacement.

    • Displacement (Vector Quantity): Displacement is the straight-line distance between the initial and final positions of an object. Displacement is a vector. It is always a straight line, regardless of the actual path taken between two locations. To draw a displacement vector, simply draw an arrow from the initial position of an object to the final position.

    • Distance (Scalar Quantity): In contrast, the total distance walked through the maze from start to finish is called distance. Distance is a measure of the length of the path taken between two positions. It is a scalar quantity, meaning it does not have a direction like displacement.

Average Velocity or Average Motion

The average velocity of an object is determined by its change in position over time. The motion can be described as the average motion over time. The average velocity from 1 to 3 can be described as:

Instantaneous Velocity

Instantaneous velocity is measured at a single point in time. It shows how fast an object is moving at an exact moment, just like a speedometer on a car shows how fast the car is moving at every moment along its path. If an object is speeding up or changing direction, then its motion is not constant in time. Instantaneous velocity varies from time to time due to acceleration. In one dimensional motion, acceleration indicates the object is either speeding up or slowing down. The instantaneous velocity is shown graphically as the slope of the tangent line on a position vs. time graph.

Students can approximate the instantaneous velocity of an object for a given time if the speed of the object is known at points in time close to that one in question. With a graph of the motion, students can select two points, one before and one after, and find the slope of the line segment between those two points. In this case, it is an approximation of the instantaneous velocity, but they can calculate it using:

velocity = (s2 - s1)/(t2 - t1)