JEE Physics Motion In A Straight Line Notes Set 01

Distance and Displacement

Distance is the total length of the path travelled and is a scalar that is never negative. Displacement is the straight line vector from the initial to the final position and can be positive, negative or zero.
The magnitude of displacement is never larger than the distance. They are equal only for motion along a straight line without turning back. A body that returns to its start has zero displacement but non-zero distance.

Speed and Velocity

Average speed = total distance / total time. Average velocity = total displacement / total time. Instantaneous velocity is the limit of the average velocity as the time interval shrinks to zero, \( v=\frac{dx}{dt} \).
If a body covers equal distances with speeds \( v_1 \) and \( v_2 \), the average speed is the harmonic mean \( \frac{2v_1v_2}{v_1+v_2} \). If it moves for equal times with those speeds, the average speed is the simple mean \( \frac{v_1+v_2}2 \).

Acceleration

\( a=\frac{dv}{dt}=v\frac{dv}{dx} \). Acceleration is positive or negative according to the chosen axis. A body speeds up when velocity and acceleration have the same sign and slows down when the signs are opposite.
A body can have zero velocity and non-zero acceleration, for example at the highest point of a vertical throw.

Equations of Motion

For constant acceleration along a straight line: \( v=u+at \), \( s=ut+\frac12at^2 \), \( v^2=u^2+2as \) and \( s=\frac{u+v}2t \).
The distance travelled in the nth second is \( s_n=u+\frac a2(2n-1) \). From rest, equal successive time intervals give distances in the ratio 1 : 3 : 5 : 7 and the time for equal successive distances follows the ratio \( 1:(\sqrt2-1):(\sqrt3-\sqrt2) \).

Calculus Method

When x is given as a function of t, differentiate once for velocity and twice for acceleration. When acceleration is given as a function of t, integrate to get the velocity and the position, using the initial conditions to fix the constants.
Examples: \( x=at^2-bt^3 \) has zero acceleration at \( t=\frac a{3b} \); \( a=kt \) gives \( v=u+\frac{kt^2}2 \).

Graphs of Motion

In a position-time graph the slope is the velocity. In a velocity-time graph the slope is the acceleration and the area under the curve is the displacement (take areas below the axis as negative; for distance add all the areas). In an acceleration-time graph the area is the change in velocity.
A horizontal v-t graph means uniform motion; a sloping straight line means uniform acceleration. A v-t graph cannot be a closed loop and a body cannot have two velocities at one time.

Motion Under Gravity

Near the earth every freely falling body has the same acceleration g (about 9.8 or 10 m/s squared) downwards, whatever its mass. Take upward as positive and use \( a=-g \).
A body thrown up with speed u rises to \( H=\frac{u^2}{2g} \) in time \( \frac ug \) and has total time of flight \( \frac{2u}g \). It returns to the starting point with the same speed u. Falling from a height h takes \( \sqrt{\frac{2h}g} \) and gives final speed \( \sqrt{2gh} \).

Air Resistance and Special Cases

If a constant air resistance acts, the retardation while rising is \( g+\frac fm \) and the acceleration while falling is \( g-\frac fm \). So the ascent is shorter than the descent and the body returns with less speed than it was thrown.
For a body released from a rising or descending balloon or lift, start with the velocity of the balloon or lift. For two bodies in free fall use relative motion: their relative acceleration is zero.

Relative Motion in One Dimension

The velocity of A relative to B is \( v_{AB}=v_A-v_B \). For bodies moving in opposite directions the relative speed is the sum of the speeds and for the same direction it is the difference.
Crossing time of two trains = (sum of lengths) / (relative speed). To overtake or to meet, divide the initial gap by the relative speed. For a walker on a moving escalator the rates add: \( \frac1t=\frac1{t_1}+\frac1{t_2} \).