JEE Physics Motion In A Straight Line Notes Set 02

Motion Along a Straight Line: Basic Terms

Position tells where a body is on a line, measured from a chosen origin. Displacement is the change in position and can be positive or negative, while distance is the total length of path travelled and is never negative. Speed is distance divided by time; velocity is displacement divided by time. Speed is a scalar and velocity is a vector, so average speed is at least as large as the magnitude of average velocity.

Average and Instantaneous Quantities

Average velocity = total displacement / total time. Average speed = total distance / total time. Instantaneous velocity is \( v = \frac{dx}{dt} \) and instantaneous acceleration is \( a = \frac{dv}{dt} = v\frac{dv}{dx} \). If a body covers equal distances at 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 at \( v_1 \) and \( v_2 \), the average speed is the arithmetic mean \( \frac{v_1+v_2}{2} \).

Equations of Uniformly Accelerated Motion

For constant acceleration a, initial velocity u, final velocity v, time t and displacement s:
\( v = u + at \)
\( s = ut + \frac{1}{2}at^2 \)
\( v^2 = u^2 + 2as \)
\( s_n = u + \frac{a}{2}(2n-1) \) (distance in the nth second)
In successive equal time intervals t, the distances covered differ by \( at^2 \).

Free Fall and Vertical Throw

Near the surface of the earth a freely falling body has constant acceleration g (about 10 m/s² in problems). For a body thrown up with speed u: time to reach the top \( \frac{u}{g} \), maximum height \( \frac{u^2}{2g} \), total time of flight \( \frac{2u}{g} \), and the body returns with the same speed u. For a body dropped from rest, the distances in successive seconds are in the ratio 1 : 3 : 5 : ... and the distance fallen is proportional to \( t^2 \).

Graphs of Straight-Line Motion

A turning point on the x-t graph (a peak or valley) means zero velocity. A graph of v against t must give one value of v for every t, so closed curves are impossible. Zero velocity at an instant does not mean zero acceleration, as at the top of a vertical throw.

Relative Motion in One Dimension

The velocity of A relative to B is \( v_{AB} = v_A - v_B \). Two bodies with the same acceleration have zero relative acceleration, so their separation changes uniformly. To find when one body overtakes another, use the relative velocity and the initial gap.

Solved Examples

Example 1. A car starts from rest with acceleration 2 m/s² and moves for 10 s. Find the distance covered.
\( s = \frac{1}{2} \times 2 \times 10^2 = 100 \) m.

Example 2. A body moves with \( x = t^3 - 6t^2 + 9t \). When is it momentarily at rest?
\( v = 3t^2 - 12t + 9 = 3(t-1)(t-3) \), so v = 0 at t = 1 s and t = 3 s.

Example 3. A body covers the first half of a distance at 30 km/h and the second half at 60 km/h. Find the average speed.
Average speed = \( \frac{2 \times 30 \times 60}{30+60} = 40 \) km/h.

Example 4. A stone is dropped from a 80 m high tower (g = 10 m/s²). How long does it take to reach the ground?
\( 80 = 5t^2 \), so t = 4 s.

Key Points to Remember

1. Use \( a = v\,dv/dx \) whenever acceleration is given as a function of position.
2. Equal distances need the harmonic mean for average speed; equal times need the arithmetic mean.
3. The area under a v-t graph is displacement, not distance, when the velocity changes sign.
4. Always check that a time obtained from algebra is physically possible (for example less than the time of flight).