Motion in a Plane
3.1 – 3.3 Scalar & Vector Classification
Q3.1 Classify as Scalar or Vector: Volume, Speed, Velocity, etc.
Scalars (Magnitude only):
Volume, Mass, Speed, Density, Number of moles, Angular frequency.
Vectors (Magnitude + Direction):
Acceleration, Velocity, Displacement, Angular velocity.
Q3.2 Pick the two scalar quantities from the list.
The scalar quantities are:
1. Work (Dot product of force and displacement, hence scalar).
2. Current (Possesses direction but follows scalar addition laws).
Q3.3 Pick the only vector quantity from the list.
The only vector quantity is:
Impulse (Defined as change in momentum, \(\Delta \vec{p} = \vec{F} \Delta t\), which is a vector).
3.4 – 3.5 Algebraic Operations & Statements
Q3.4 Meaningful algebraic operations.
(a) Adding any two scalars: No. Only scalars of same dimensions can be added (e.g., Mass + Mass).
(b) Adding a scalar to a vector: No. Meaningless to add directionless quantity to one with direction.
(c) Multiplying vector by scalar: Yes. E.g., \(\vec{F} = m \vec{a}\).
(d) Multiplying any two scalars: Yes. E.g., Power = Work / Time.
(e) Adding any two vectors: No. Must be of same dimension (Velocity + Velocity).
(f) Adding vector component to same vector: Yes. A component is a vector of the same dimension.
Q3.5 True or False with reasons.
(a) Magnitude of a vector is always a scalar: True. It is a pure number with units.
(b) Each component of a vector is always a scalar: False. Components are vectors along specific axes.
(c) Total path length equals displacement magnitude: False. Only true for 1D motion without reversing. In general, Path \(\ge\) |Displacement|.
(d) Avg speed \(\ge\) |Avg velocity|: True. Because total path length \(\ge\) displacement magnitude.
(e) Three vectors not in a plane can never add to zero: True. The resultant of two vectors lies in their plane; it cannot cancel a third vector outside that plane.
3.6 – 3.7 Vector Inequalities
Q3.6 Establish vector inequalities geometrically.
(a) \(|\vec{a}+\vec{b}| \le |\vec{a}| + |\vec{b}|\):
According to triangle inequality, the length of one side is always less than the sum of the other two sides. Equality holds if vectors are collinear and in same direction.
(b) \(|\vec{a}+\vec{b}| \ge ||\vec{a}| – |\vec{b}||\):
One side of a triangle is always greater than the difference of the other two sides.
(c) & (d) Similar logic for subtraction \(\vec{a} – \vec{b}\).
Replace \(\vec{b}\) with \(-\vec{b}\) in the triangle law.
Q3.7 Given \(\vec{a} + \vec{b} + \vec{c} + \vec{d} = 0\), which statements are correct?
(a) Incorrect. They form a closed polygon; individual vectors need not be null.
(b) Correct. \(\vec{a} + \vec{c} = -(\vec{b} + \vec{d})\). Magnitudes must be equal.
(c) Correct. Length of one side of a polygon is always less than sum of other sides.
(d) Correct. Since \(\vec{a} + \vec{d} = -(\vec{b} + \vec{c})\), the resultant of \(\vec{b}+\vec{c}\) must be collinear with the resultant of \(\vec{a}+\vec{d}\) to cancel out.
3.8 – 3.10 Displacement Problems
Q3.8 Three girls skating on circular ice.
Magnitude of Displacement:
For all girls, start is P and end is Q (diametrically opposite).
Displacement vector magnitude = Diameter = \(2R = 2 \times 200 = 400 \text{ m}\).
Equality to Path Length:
Only for Girl B, who skates straight across the center. For A and C, path > 400m.
Q3.9 Cyclist in Circular Park.
Path: Center O \(\to\) P \(\to\) Circumference \(\to\) Q \(\to\) Center O.
Time = 10 min = 1/6 hr. Radius R = 1 km.
(a) Net Displacement: Start = End = O. Zero.
(b) Average Velocity: Disp/Time = Zero.
(c) Average Speed:
Path = \(OP + \text{Arc } PQ + QO = 1 + \frac{\pi(1)}{2} + 1 \approx 3.57 \text{ km}\).
Speed = \(3.57 / (1/6) = 21.42 \text{ km/h}\).
Q3.10 Motorist turning 60° every 500m.
Path is a regular hexagon (60° exterior angle). Side = 500m.
3rd Turn: Opposite end. Disp = \(2 \times 500 = 1000 \text{ m}\). Path = 1500m.
6th Turn: Back to start. Disp = 0. Path = 3000m.
8th Turn: Equivalent to 2nd turn vertex (C).
Disp AC (diagonal) = \(500\sqrt{3} \approx 866 \text{ m}\).
Path = \(8 \times 500 = 4000 \text{ m}\).
3.11 – 3.13 General Kinematics
Q3.11 Cab ride to hotel.
Data: Displacement = 10km (straight line). Path = 23km. Time = 28 min = 28/60 h.
(a) Average Speed: Path/Time = \(23 / (28/60) = 49.3 \text{ km/h}\).
(b) Avg Velocity Mag: Disp/Time = \(10 / (28/60) = 21.4 \text{ km/h}\).
Equality: No, path length > displacement magnitude.
Q3.12 Maximum range in hall with ceiling.
Given \(H_{max} = 25 \text{ m}\), \(u = 40 \text{ m/s}\).
\(\sin^2 \theta = 0.30625 \Rightarrow \sin \theta = 0.553\).
\(\cos \theta = \sqrt{1 – 0.30625} = 0.833\).
Range: \(R = \frac{u^2 (2 \sin \theta \cos \theta)}{g} = \frac{1600 \times 2 \times 0.553 \times 0.833}{9.8} = 150.5 \text{ m}\).
Q3.13 Cricketer throwing ball.
Max horizontal range \(R_{max} = \frac{u^2}{g} = 100 \text{ m}\).
For max height, throw vertically (\(90^\circ\)).
\(H_{max} = \frac{u^2}{2g} = \frac{1}{2} \left(\frac{u^2}{g}\right) = \frac{100}{2} = 50 \text{ m}\).
3.14 – 3.16 Circular Motion
Q3.14 Stone whirled in horizontal circle.
Radius \(R = 0.8 \text{ m}\). Frequency \(\nu = 14/25 \text{ Hz}\).
Acceleration \(a = \omega^2 R = \left(\frac{88}{25}\right)^2 \times 0.8 \approx 9.9 \text{ m/s}^2\).
Direction: Towards the center (Centripetal).
Q3.15 Aircraft loop.
\(R = 1000 \text{ m}\). \(v = 900 \text{ km/h} = 250 \text{ m/s}\).
Ratio \(a_c : g = 62.5 : 9.8 \approx 6.4\).
Q3.16 Circular motion True/False.
(a) Net acceleration always towards center: False. Only in Uniform Circular Motion. Non-uniform has tangential component.
(b) Velocity vector always along tangent: True.
(c) Avg acceleration over one cycle is null: True. \(\Delta \vec{v} = 0\) over full cycle.
3.17 – 3.22 Vector Problems
Q3.17 Position vector differentiation.
\(\vec{r} = 3t\hat{i} – 2t^2\hat{j} + 4\hat{k}\).
(a) v and a:
\(\vec{v} = \frac{d\vec{r}}{dt} = 3\hat{i} – 4t\hat{j}\).
\(\vec{a} = \frac{d\vec{v}}{dt} = -4\hat{j}\).
(b) At t=2s:
\(\vec{v} = 3\hat{i} – 8\hat{j}\).
Mag = \(\sqrt{3^2 + (-8)^2} = \sqrt{73} \approx 8.54 \text{ m/s}\).
Dir = \(\tan^{-1}(-8/3) \approx -69.5^\circ\) (below x-axis).
Q3.18 Particle in xy plane.
\(\vec{u} = 10\hat{j}\). \(\vec{a} = 8\hat{i} + 2\hat{j}\).
(a) Time for x=16m:
\(x = u_x t + \frac{1}{2}a_x t^2 \Rightarrow 16 = 0 + \frac{1}{2}(8)t^2 \Rightarrow t = 2 \text{ s}\).
y-coordinate: \(y = 10(2) + \frac{1}{2}(2)(2^2) = 24 \text{ m}\).
(b) Speed:
\(v_x = 0 + 8(2) = 16\). \(v_y = 10 + 2(2) = 14\).
Speed = \(\sqrt{16^2+14^2} \approx 21.26 \text{ m/s}\).
Q3.19 Magnitude and Components.
\(\hat{i} + \hat{j}\): Mag = \(\sqrt{1^2+1^2}=\sqrt{2}\). Dir = \(45^\circ\) to x-axis.
\(\hat{i} – \hat{j}\): Mag = \(\sqrt{2}\). Dir = \(-45^\circ\) to x-axis.
Components of \(\vec{A} = 2\hat{i} + 3\hat{j}\):
Along \(\hat{i}+\hat{j}\) (direction \(\hat{n} = \frac{\hat{i}+\hat{j}}{\sqrt{2}}\)): \(\vec{A} \cdot \hat{n} = \frac{2+3}{\sqrt{2}} = \frac{5}{\sqrt{2}}\).
Along \(\hat{i}-\hat{j}\): \(\vec{A} \cdot \hat{n}’ = \frac{2-3}{\sqrt{2}} = \frac{-1}{\sqrt{2}}\).
Q3.20 Arbitrary motion relations.
(a) False. Average velocity is not simple average of initial and final unless acceleration is constant.
(b) True. Definition of average velocity (Displacement / Time).
(c) False. Only valid for constant acceleration.
(d) False. Only valid for constant acceleration.
(e) True. Definition of average acceleration.
Q3.21 Scalar quantity properties.
(a) Conserved: False. Kinetic energy (scalar) is not conserved in inelastic collisions.
(b) Never negative: False. Temperature can be -20°C.
(c) Dimensionless: False. Mass has dimensions [M].
(d) Constant in space: False. Gravitational potential varies with position.
(e) Same for all observers: True. Scalar value (like Mass) is invariant under rotation of axes.
Q3.22 Aircraft Speed (Parallax).
Height \(h = 3400 \text{ m}\). Angle \(\theta = 30^\circ\). Time \(t = 10 \text{ s}\).
Let \(O\) be observation point. \(A\) and \(B\) be positions.
Distance \(AB\) (Base of triangle) is \(x\).
\(\tan(\theta/2) = \frac{x/2}{h} \Rightarrow x = 2h \tan(15^\circ)\).
\(x = 2 \times 3400 \times 0.2679 = 1822 \text{ m}\).