Motion In A Straight Line
Motion
Motion is change in position of an object with time.Motion of object along a straight line is called rectilinear motion. Examples include flying kite, moving train, earth's rotation etc.
Frame of Reference
In order to know the change in position of an object, a reference point is required. Point O in the figure is the reference point or Origin and together with three axes, this system is called the coordinate system. A coordinate system with time frame is called frame of reference.
- Objects changing positions with time with respect to the frame of reference are in motion while those which do not change position are at rest.
- For a moving car, for the frame of reference outside the car, it appears moving. While for the frame of reference inside the car, the car appears stationary.
Motion along a straight line
Motion along a straight line is described using only X-axis of the coordinate system.
Path Length (Distance) Vs. Displacement
Path Length: It is the distance between two points along a straight line. It is scalar quantity.
Displacement: It is the change in position in a particular time interval. It is vector quantity. Change is position is usually denoted by Δx (x_{2}-x_{1}) and change in time is denoted by Δt (t_{2}-t_{1}).
For the above example, if a person goes from home (O) to school (x_{2}) and comes back from school to Park (x_{1}), then
Path length(Home to School and School to Park) = Ox_{2} + x_{2}x_{1} = (+80) + (+60) = +140m. This is always positive.
Displacement(Home to Park) = Ox_{2} - x_{2}x_{1} = +80 – (+60) = +20m. This can be positive as well as negative. The negative sign indicates the direction.
- Magnitude of Displacement may or may not be equal to the path length.
- For a non-zero path length, displacement can be 0 (case where an object returns to origin).
Position-Time, Velocity-Time and Acceleration-Time Graph
Criteria | P-T Graph | V-T Graph | A-T Graph |
X and Y axis | Time and Position | Time and Velocity | Time and Acceleration |
Slope | It represents velocity of an object | It represents acceleration of an object. | It represents the jerk or push of a moving object. |
Straight slope | Uniform velocity | Uniform acceleration | Uniform jerk |
Curvy Slope | Change in velocity | Change in acceleration | Change in the amount of push/jerk |
Average Velocity and Average Speed
Criteria | Average Velocity | Average Speed |
Definition | Change in position or displacement divided by time interval. | Total path length travelled divided by total time interval regardless of direction. |
Formula | Avg speed = Total path length/Total time interval | |
Scalar or Vector | Vector | Scalar |
Sign | Can be positive or negative | Always positive |
Unit | m/s | m/s |
Instantaneous Velocity and Instantaneous Speed
Instantaneous velocity describes how fast an object is moving at different instants of time in a given time interval. It is also defined as average velocity for an infinitely small time interval.
Here lim is taking operation of taking limit with time tending towards 0 or infinitely small.
dx/dt is differential coefficient – Rate of change of position with respect to time at an instant.
Instantaneous speed is the magnitude of velocity. Instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.
Acceleration
Acceleration is rate of change of velocity with time. It is denoted by 'a' and the SI unit is m/s^{2}.
Average acceleration is change of velocity over a time interval.
Here v_{1} and v_{2 }are instantaneous velocities at time t_{1} and t_{2}.
- Acceleration can be positive (increasing velocity) or negative (decreasing velocity).
- Instantaneous acceleration is acceleration at different instants of time. Acceleration at an instant is slope of tangent to the v-t curve at that instant.
- For a velocity v_{0} at time t=0, the velocity v at time t will be, v = v_{0 }+ a Area under v-t curve represents displacement over given time interval.
- Acceleration and velocity cannot change values abruptly. The changes are continuous.
Kinematic equations for uniformly accelerated motion
There are 3 kinematic equations of rectilinear motion for constant acceleration
Position of object at time t=0 is 0 | Position of object at time t=0 is x_{0} |
v = v_{0 }+ at | v = v_{0 }+ at |
x = v_{0}t + ½ at^{2} | x = x_{0}+ v_{0}t + ½ at^{2} |
v^{2} = v_{0}^{2} + 2ax | v^{2} = v_{0}^{2} + 2a(x-x_{0}) |
Relative Velocity
This is the velocity of an object relative to some other object which might be stationary, moving slowly, moving with same velocity, moving with higher velocity or moving in opposite direction.
If initial position of two objects A and B are x_{A} (0) and x_{B} (0), the position at time t will be,
- x_{A} (t) = x_{A} (0) + v_{A} t
- x_{B} (t) = x_{B} (0) + v_{B} t
- Displacement from object A to B, [ x_{B} (0) - x_{A} (0) ] + (v_{B} -v_{A})t
- Velocity of B relative to A = v_{BA} = v_{B} – v_{A}
- Velocity of A relative to B = v_{AB} = v_{A} – v_{B}
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