Motion

Displacement

Consider the motion of a point that is constrained to move in a straight line.

To distinguish the two points we introduce the concept of displacement. The displacement of a point from O will be a quantity whose numerical value will be the distance and its sign will tell us if the point is to the right or left of O.

Displacement

We will use the symbol x for displacement in a straight line and s for distance. Displacement, being a vector is represented graphically by an arrow that begins at O and ends at the point of interest.

We will use the capital letter S to stand for the total distance travelled, and Δx for the change in displacement. The change in displacement is defined by

Δx = final displacement – initial displacement

Example questions

Speed

If an object covers a total distance S in a total time T, the average speed of the objects is defined by

Velocity

Average speed is a positive quantity that does not take into account the direction in which the object moves. To do that we introduce the concept of velocity. The average velocity for a motion is defined as the change in displacement of the object divided by the total time taken.

Example questions

Uniform motionMotion with uniform velocity means motion in which the velocity is constant. This implies that the displacement changes by equal amounts in equal intervals of time (no matter how small or large). Let us take the interval o time from t = 0 to time t.

If the displacement at t = 0 is x0 and the displacement at time t is x, then it follows that

v = Δx/Δt = (x-x0)/(t-t0)

x = x0 + vt

Example questions

Frames of reference

There is no absolute meaning to the statement ‘being at rest’. No experiment can be performed the result of which will be to let observers know that they are moving with constant velocity and that they are not at rest.

An observer who uses measuring tapes and stopwatches to observe and record motion is called a frame of reference.

Relative velocity

This introduces the concept of relative velocity. Let two observers P and Q have velocities vP and vQ as measured by the same frame of reference. Then the relative velocity of P with respect to Q is simply

Example questions

Graphs for uniform motion

Graphs for uniform motion

Example question

…more questions

Acceleration

To treat situations in which velocity is not constant we need to define acceleration a. If the velocity changes by Δv in a very short interval of time Δt then

a = Δv/Δt

is the definition of the instantaneous acceleration.

When acceleration is constant, the motion is called uniformly accelerated motion.

AccelerationFor motion in a straight line, positive acceleration means that the velocity is increasing whereas negative acceleration implies a decreasing velocity.

Acceleration due to gravity

When an object is dropped or thrown, there is an acceleration that acts on all objects and has the same magnitude for all bodies independently of their mass.

On earth the magnitude of this acceleration is about 9.8 m/s2.

Example question

Graphs for uniformly accelerated motion

Displacement

Even in the case of accelerated motion, the change in displacement is the area under the velocity-time graph.

Example questions

Graphs of acceleration versus timeIn a graph of acceleration versus time the area under the graph gives the change in velocitiy.

Graphs of displacement versus time

In motion with constant acceleration, a graph of displacement versus time is a parabola. Drawing the tangent at a point on the curve and finding the slope of the tangent gives the velocity at that point.