Motion in a Plane — JEE Mains Physics

Formulas (3)

• Resultant of two vectors at an angle · Resultant of two vectors
  $$R = \sqrt{A^{2} + B^{2} + 2AB\cos\theta} \qquad \tan\alpha = \frac{B\sin\theta}{A + B\cos\theta} \qquad |\vec{A} + \vec{B}| = 2A\cos\frac{\theta}{2}\ (A = B)$$
• Resolving vectors into components · Components and magnitude
  $$A_x = A\cos\theta,\ A_y = A\sin\theta \qquad A = \sqrt{A_x^{2} + A_y^{2} + A_z^{2}} \qquad \vec{A}\ \text{along}\ \vec{B} = A\hat{B}$$
• Dot product, cross product and projection · Products and projection
  $$\vec{A}\cdot\vec{B} = AB\cos\theta \qquad |\vec{A}\times\vec{B}| = AB\sin\theta \qquad \text{proj}_{B}\vec{A} = \frac{\vec{A}\cdot\vec{B}}{|\vec{B}|}$$

Watch out for (6)

• Adding magnitudes
  → Resultant of two vectors at an angle
• Perpendicular to A, not to B
  → Resultant of two vectors at an angle
• Angle with the y-axis
  → Resolving vectors into components
• Losing the quadrant
  → Resolving vectors into components
• Dividing the projection by the wrong length
  → Dot product, cross product and projection
• Order matters in a cross product
  → Dot product, cross product and projection

Formulas (3)

• Position, velocity and acceleration vectors · Motion in two dimensions
  $$\vec{v} = \frac{d\vec{r}}{dt},\ \vec{a} = \frac{d\vec{v}}{dt} \qquad \vec{r} = \vec{r}_0 + \vec{u}t + \tfrac{1}{2}\vec{a}t^{2} \qquad \tan\theta = \frac{v_y}{v_x}$$
• Crossing a river · Crossing time and drift
  $$t_{\min} = \frac{d}{v} \qquad x_{drift} = \frac{ud}{v} \qquad \sin\alpha = \frac{u}{v},\ t = \frac{d}{\sqrt{v^{2} - u^{2}}}$$
• Relative velocity and change of frame · Relative velocity
  $$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B \qquad \tan\theta_{umbrella} = \frac{v_{man}}{v_{rain}}\ (\text{vertical rain})$$

Watch out for (6)

• Quoting an angle without its axis
  → Position, velocity and acceleration vectors
• Force along the motion
  → Position, velocity and acceleration vectors
• Dividing the width by the resultant speed
  → Crossing a river
• Least time is not shortest path
  → Crossing a river
• Subtracting in the wrong order
  → Relative velocity and change of frame
• Taking a released object to start from rest
  → Relative velocity and change of frame

Formulas (3)

• Time of flight, maximum height and range · Projectile on level ground
  $$T = \frac{2u\sin\theta}{g} \qquad H = \frac{u^{2}\sin^{2}\theta}{2g} \qquad R = \frac{u^{2}\sin 2\theta}{g}$$
• Maximum range and the range-height relation · Greatest range; range against height
  $$R_{\max} = \frac{u^{2}}{g}\ (45^{\circ}) \qquad \frac{R}{H} = \frac{4}{\tan\theta}$$
• Complementary angles give the same range · Two angles, one range
  $$R = 4\sqrt{H_1H_2} \qquad T_1T_2 = \frac{2R}{g} \qquad H_1 + H_2 = \frac{u^{2}}{2g}$$

Watch out for (6)

• sin 2θ in the range, sin²θ in the height
  → Time of flight, maximum height and range
• Time of flight does not use sin 2θ
  → Time of flight, maximum height and range
• Greatest height equal to greatest range
  → Maximum range and the range-height relation
• 45° is only for level ground
  → Maximum range and the range-height relation
• Equal range does not mean equal height
  → Complementary angles give the same range
• Adding the angles to 180°
  → Complementary angles give the same range

Formulas (3)

• Velocity and kinetic energy during the flight · Velocity during the flight
  $$v_x = u\cos\theta,\ v_y = u\sin\theta - gt \qquad K_{top} = K\cos^{2}\theta \qquad h = \tfrac{1}{2}gt_1t_2$$
• Equation of the trajectory · Path of a projectile
  $$y = x\tan\theta - \frac{gx^{2}}{2u^{2}\cos^{2}\theta} \qquad y = \alpha x - \beta x^{2}:\ R = \frac{\alpha}{\beta},\ H = \frac{\alpha^{2}}{4\beta}$$
• Thrown horizontally from a height · Horizontal projection from height h
  $$t = \sqrt{\frac{2h}{g}} \qquad x = u\sqrt{\frac{2h}{g}} \qquad v = \sqrt{u^{2} + 2gh}$$

Watch out for (6)

• Zero speed at the top
  → Velocity and kinetic energy during the flight
• cos θ instead of cos²θ
  → Velocity and kinetic energy during the flight
• Dropping the cos²θ
  → Equation of the trajectory
• The top is at x = R/2
  → Equation of the trajectory
• Giving the horizontally thrown body a vertical speed
  → Thrown horizontally from a height
• Horizontal distance is not displacement
  → Thrown horizontally from a height

Formulas (2)

• Distance, displacement and average velocity on a circle · Arc and chord
  $$s = r\theta \qquad |\Delta\vec{r}| = 2r\sin\frac{\theta}{2} \qquad \frac{v}{v_{avg}} = \frac{\theta}{2\sin(\theta/2)}$$
• Centripetal and tangential acceleration · Circular motion
  $$v = \omega r \qquad a_c = \frac{v^{2}}{r} = \omega^{2}r \qquad a = \sqrt{a_t^{2} + a_c^{2}},\ a_t = \frac{dv}{dt}$$

Watch out for (5)

• Average velocity from the arc
  → Distance, displacement and average velocity on a circle
• Degrees in s = rθ
  → Distance, displacement and average velocity on a circle
• Constant speed, zero acceleration
  → Centripetal and tangential acceleration
• Forgetting to convert rpm
  → Centripetal and tangential acceleration
• Adding the two accelerations directly
  → Centripetal and tangential acceleration

Formulas (3)

• Friction and banking on a curve · Roads and turntables
  $$v_{\max} = \sqrt{\mu rg} \qquad \tan\theta = \frac{v^{2}}{rg} \qquad v_{\max}^{2} = rg\,\frac{\tan\theta + \mu}{1 - \mu\tan\theta}$$
• String, spring and wall in a horizontal circle · Horizontal circles
  $$T = m\omega^{2}L \qquad kx = m\omega^{2}(l_0 + x) \qquad \tan\theta = \frac{v^{2}}{rg}\ (\text{conical pendulum})$$
• Motion in a vertical circle · Vertical circle on a string
  $$T_b = \frac{mv_b^{2}}{r} + mg,\ T_t = \frac{mv_t^{2}}{r} - mg \qquad v_b^{2} = v_t^{2} + 4gr \qquad v_b \ge \sqrt{5gr}$$

Watch out for (6)

• Putting the mass into the safe speed
  → Friction and banking on a curve
• Friction with the wrong sign on a banked road
  → Friction and banking on a curve
• Using the natural length as the radius
  → String, spring and wall in a horizontal circle
• Radius or string length in a conical pendulum
  → String, spring and wall in a horizontal circle
• Zero speed at the top
  → Motion in a vertical circle
• Change in velocity as a change in speed
  → Motion in a vertical circle