Kinematics Concepts Page - 4

Definition

Instantaneous velocity

Instantaneous velocity is defined as the rate of change of displacement with time,where the period of time is narrowed such that it reaches zero.
v⃗ =ds⃗ dt
Note:
1. Direction of instantaneous velocity at any time gives the direction of motion of particle at that point of time.
2. Magnitude of instantaneous velocity equals the instantaneous speed. This happens because for an infinitesimally small time interval, motion of a particle can be approximated to be uniform.

Example

Average Velocity

For a body moving with uniform acceleration a, initial and final velocities in a time interval t are u and v respectively.

Then, its average velocity in the time interval t$t$ will be given by:
S=ut+12at2
v=u+at
Average velocity =Total displacement(s)Total time(t)
=ut+12at2t=u+12at
Substitute, u=v−at in the above expression,
We get average velocity =v−at+12at=v−12at

Definition

Uniform acceleration

A body under constant acceleration is called as uniformly accelerated motion. This happens when a constant external force is applied on the body.
Note:
Direction of acceleration plays a role in the change of velocity. However, direction of acceleration of a body may not be same as the direction of its motion.

Definition

Define and calculate instantaneous acceleration

When the object is moving with variable acceleration, then the object possesses different acceleration at different instant. The acceleration of the object at different instant of time or at given time of motion, is called instantaneous acceleration.

Definition

Average Acceleration

Average acceleration is the change in velocity divided by an elapsed time. For instance, if the velocity of a marble increases from 0 to 60 cm/s in 3 seconds, its average acceleration would be 20 cm/s2. This means that the marble's velocity will increase by 20 cm/s every second.

Formula

Understand Acceleration as Second Derivative of Displacement

a⃗ =d2x⃗ dt2
Where x$x$ is a displacement vector.

Definition

Acceleration as a rate of change of velocity with respect to displacement

Acceleration is defined as a=dvdt
∴a=dxdvdxdt
a=vdvdx

Example:

If the velocity of a particle is given by v=(180−16x)1/2m/s, then find its acceleration.

Solution:

a=dvdt=dxdtdvdx=vdvdx

v=(180−16x)12v2=180−16x

Differentiate both sides w.r.t x, we get:

2vdvdx=−16vdvdx=a=−8 m/s2

Formula

Definition of Jerk

Rate of change of acceleration is defined as jerk.
jerk = da⃗ dt

Result

Distance in nth second(distance-time graph)

Distance traveled in nth second can be found using distance time-graph. In the given figure, distance after 21 s is d2=21 m and after 20 s is d1=20 m. Hence, distance traveled in 21st sec is d2−d1=1 m.

Definition

Average velocity from a displacement-time graph

Average velocity equals the ratio of net displacement and the time taken.
Example:
In the given plot, for t∈(0,10),
Displacement s=−4−0=−4
Hence, average velocity is given by v=s/t=−4/10=−0.4m/s

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