JEE Physics Motion In A Straight Line Notes Set 03

Describing Motion in a Straight Line

1. Distance is the length of the path and is never negative. Displacement is the change in position and can be positive, negative or zero. Speed is distance over time, and velocity is displacement over time.
2. Average velocity \(= \frac{\Delta x}{\Delta t}\). Instantaneous velocity \(v = \frac{dx}{dt}\) and acceleration \(a = \frac{dv}{dt} = v\frac{dv}{dx}\).
3. A body can have zero velocity and non-zero acceleration at an instant, as at the top of a vertical throw, and zero acceleration while moving, as in uniform motion.

Equations of Uniformly Accelerated Motion

For constant acceleration: \(v = u + at\), \(s = ut + \frac{1}{2}at^2\), \(v^2 = u^2 + 2as\), and the distance in the \(n\)th second \(s_n = u + \frac{a}{2}(2n - 1)\). If the acceleration varies with time, for example \(a = kt\), integrate: \(v = u + \frac{1}{2}kt^2\). Do not use the constant-acceleration formulae there.

Free Fall and Vertical Throw

1. Take \(a = -g\) with upward positive. Time to the top \(= \frac{u}{g}\), maximum height \(= \frac{u^2}{2g}\), time of flight \(= \frac{2u}{g}\).
2. A body passes any point at the same speed going up and coming down. If it passes a height at times \(t_1\) and \(t_2\), the time to the top is \(\frac{t_1 + t_2}{2}\).
3. From rest, distances covered in successive equal times are in the ratio \(1 : 3 : 5 : \ldots\). Times for successive equal heights are \(1 : (\sqrt{2} - 1) : (\sqrt{3} - \sqrt{2}) : \ldots\).
4. A released body has acceleration \(g\) downward irrespective of the motion of its carrier, such as a rising lift or balloon.

Graphs

1. A steeper position-time line means greater speed. A curved S shape has its maximum speed at the point of inflection.
2. For a velocity-displacement graph use \(a = v\frac{dv}{dx}\). A straight rising line gives acceleration proportional to \(x\).
3. Distance is the area ignoring signs, and displacement is the area with signs.

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 speeds add, and in the same direction they subtract. When two trains cross completely, the relative distance is the sum of the lengths. For a boat in a river, downstream speed is \(v + u\) and upstream speed is \(v - u\), so the still-water time is the harmonic mean of the two journey times.

Worked Examples

1. A car starts from rest with acceleration \(2\ \text{m s}^{-2}\) for \(5\ \text{s}\), then moves uniformly for \(10\ \text{s}\), and finally stops with retardation \(5\ \text{m s}^{-2}\). Maximum speed \(= 10\ \text{m s}^{-1}\), distances \(25\), \(100\) and \(10\ \text{m}\), so the total distance is \(135\ \text{m}\) in \(5 + 10 + 2 = 17\ \text{s}\).
2. A particle has \(x = t^3 - 3t^2\). Then \(v = 3t^2 - 6t\) and \(a = 6t - 6\). The particle is momentarily at rest at \(t = 0\) and \(t = 2\ \text{s}\), and its acceleration is zero at \(t = 1\ \text{s}\) where \(v = -3\ \text{m s}^{-1}\).
3. A ball is thrown up at \(30\ \text{m s}^{-1}\). It reaches \(\frac{900}{20} = 45\ \text{m}\) after \(3\ \text{s}\), and returns after \(6\ \text{s}\) with the same speed of \(30\ \text{m s}^{-1}\) downward (\(g = 10\ \text{m s}^{-2}\)).
4. A bullet of speed \(100\ \text{m s}^{-1}\) loses half its speed passing through a plank. Using \(v^2 = u^2 - 2as\) the plank gives \(2as = 7500\), so the bullet can penetrate \(\frac{1}{3}\) of a further plank thickness before stopping.

Common Mistakes

1. Using \(v = u + at\) when the acceleration is not constant.
2. Confusing average speed with the magnitude of average velocity.
3. Forgetting that areas under a v-t graph below the axis subtract from displacement but add to distance.
4. Treating zero velocity as zero acceleration.