JEE Physics Motion In A Plane Notes Set 03

Vectors in Motion in a Plane

1. Position, velocity and acceleration are vectors. Resolve along perpendicular axes and treat each axis separately. \(\vec v = \frac{d\vec r}{dt}\) and \(\vec a = \frac{d\vec v}{dt}\).
2. For coordinates \(x(t)\) and \(y(t)\), differentiate each. Speed \(= \sqrt{v_x^2 + v_y^2}\). A term linear in \(t\) in a coordinate gives zero acceleration in that direction.
3. If \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) are given in terms of \(x\) and \(y\), divide to get \(\frac{dy}{dx}\) and integrate to find the path.

Projectile Motion

Horizontal and vertical motions are independent. With speed \(u\) at angle \(\theta\) above the horizontal and the landing on the level of projection:

1. Angles \(\theta\) and \(90^\circ - \theta\) give the same range. \(R = 4H\cot\theta\), and \(\tan\theta = \frac{4H}{R}\).
2. For a path \(y = ax - bx^2\), the angle of projection is \(\tan^{-1}a\) and the maximum height is \(\frac{a^2}{4b}\).
3. A horizontal acceleration (such as wind) does not alter the time of flight or the height, but it changes the range by \(\frac{1}{2}a_xT^2\).
4. A body fired horizontally from a height \(h\) takes \(\sqrt{\frac{2h}{g}}\) to land, whatever its horizontal speed. A body dropped from a moving carrier keeps the carrier's horizontal speed.
5. Average velocity from the start to the top is \(\frac{v}{2}\sqrt{1 + 3\cos^2\theta}\).

Relative Velocity, Rain and River

1. \(\vec v_{AB} = \vec v_A - \vec v_B\).
2. Rain and man: rain appears vertical to a man when his velocity equals the horizontal component of the rain's velocity. Otherwise he holds the umbrella tilted along the relative velocity of the rain.
3. Boat in a river (boat speed \(v\), river speed \(u\), width \(d\)): minimum time \(= \frac{d}{v}\) with the boat pointing straight across, and drift \(= \frac{ud}{v}\).
4. Shortest path (zero drift, needs \(v > u\)): steer upstream at angle \(\theta\) from the perpendicular with \(\sin\theta = \frac{u}{v}\), or at angle \(\cos^{-1}\left(-\frac{u}{v}\right)\) from the flow direction. The crossing time is \(\frac{d}{\sqrt{v^2 - u^2}}\).
5. If \(v < u\), the minimum drift is \(d\,\frac{\sqrt{u^2 - v^2}}{v}\).

Worked Examples

1. A ball is thrown at \(20\ \text{m s}^{-1}\) at \(30^\circ\). \(T = \frac{2\times 20\times 0.5}{10} = 2\ \text{s}\), \(H = \frac{400\times 0.25}{20} = 5\ \text{m}\), \(R = \frac{400\times 0.866}{10} = 34.6\ \text{m}\).
2. A boat with speed \(10\ \text{km h}^{-1}\) crosses a river \(2\ \text{km}\) wide flowing at \(6\ \text{km h}^{-1}\) by the shortest path. Crossing speed \(= \sqrt{100 - 36} = 8\ \text{km h}^{-1}\), time \(= 0.25\ \text{h} = 15\ \text{min}\).
3. A particle moves with \(x = 3t\) and \(y = 4t - 5t^2\). Velocity at \(t = 0\) is \(5\ \text{m s}^{-1}\) at \(\tan^{-1}\frac{4}{3}\) to the x-axis, acceleration is \(10\ \text{m s}^{-2}\) downward, and the path is \(y = \frac{4x}{3} - \frac{5x^2}{9}\).
4. Two projectiles with the same speed and angles \(\theta\) and \(90^\circ - \theta\) have equal ranges, and their times of flight are in the ratio \(\tan\theta : 1\).

Common Mistakes

1. Using the range formula when the landing level differs from the launch level.
2. Mixing up the angle with the horizontal and the angle with the vertical when resolving a velocity.
3. Adding vector magnitudes instead of components.
4. Assuming that a river's current changes the time to cross when the boat points straight across. It changes only the drift.