Practice Questions

Justify why, in projectile motion, the horizontal component of velocity remains constant while the vertical component changes.

Define a scalar quantity and provide two examples from the source text.

Evaluate under what specific condition the magnitude of the displacement vector of a particle is exactly equal to the path length it has traveled.

Identify which of the following physical quantities are scalars and which are vectors: volume, acceleration, speed, displacement.

Define projectile motion.

Examine why an object in uniform circular motion is considered to be accelerating, even though its speed is constant.

Contrast two vectors $\mathbf{A}$ and $\mathbf{B}$ that have the same magnitude but are in opposite directions. Are they equal?

Define uniform circular motion.

Define a vector quantity and list two examples mentioned in the text.

A particle's velocity changes from $\mathbf{v}_1 = (3\hat{\mathbf{i}} + 4\hat{\mathbf{j}}) \text{ m/s}$ to $\mathbf{v}_2 = (5\hat{\mathbf{i}} - 2\hat{\mathbf{j}}) \text{ m/s}$. Calculate the change in velocity vector, $\Delta\mathbf{v}$.

A football is kicked with an initial velocity of $25 \text{ m/s}$ at an angle of $30^\circ$ with the horizontal. Calculate the total time the football remains in the air before hitting the ground again. (Assume $g = 9.8 \text{ m/s}^2$)

Given two vectors $\mathbf{A} = 2\hat{\mathbf{i}} + 3\hat{\mathbf{j}}$ and $\mathbf{B} = -4\hat{\mathbf{i}} + 5\hat{\mathbf{j}}$, calculate their resultant vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$ and find its magnitude.

Justify why an object in uniform circular motion is considered to be accelerating, even though its speed is constant. Describe the direction of this acceleration.

A fellow student suggests adding the mass of an object (a scalar, e.g., $5 \text{ kg}$) to its velocity vector (e.g., $10\hat{\mathbf{i}} \text{ m/s}$). Critique this algebraic operation. Is it physically meaningful? Justify your reasoning.

A cricket ball is thrown with an initial speed of $30 \text{ m/s}$ at an angle of $45^\circ$ to the horizontal. Calculate the maximum height reached by the ball. (Use $g = 9.8 \text{ m/s}^2$)

Calculate the vector that must be added to the vector $\mathbf{A} = 5\hat{\mathbf{i}} - 7\hat{\mathbf{j}}$ so that the resultant is a unit vector along the positive y-direction.

Compare the horizontal ranges of two projectiles launched with the same initial speed $v_o$ but at two different projection angles, $\theta_1 = 30^\circ$ and $\theta_2 = 60^\circ$. Analyze the result and provide a general conclusion.

Propose a physical situation, other than an object's displacement, where the resultant vector is a null vector.

For a projectile launched from the origin, summarize the formulas for (a) the time taken to reach maximum height ($t_m$), (b) the maximum height ($h_m$), and (c) the horizontal range ($R$).

Recall the formula for centripetal acceleration, $a_c$, in terms of the object's linear speed $v$ and the radius $R$ of the circular path.

Analyze why a component of a vector can be negative, even though the magnitude of the vector is always positive.

A student claims that if the magnitude of the resultant of two vectors $\mathbf{A}$ and $\mathbf{B}$ is equal to the sum of their individual magnitudes, i.e., $|\mathbf{A} + \mathbf{B}| = |\mathbf{A}| + |\mathbf{B}|$, then the two vectors must be parallel and point in the same direction. Justify this claim mathematically.

Critique the statement: 'An object in uniform circular motion has a constant acceleration vector'.

What is a unit vector? Recall the expression for a unit vector $\hat{\mathbf{n}}$ in the direction of a given vector $\mathbf{A}$.

A vector is given by $\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}}$. Explain the steps to find its magnitude $A$ and its direction $\theta$ with respect to the positive x-axis.

Recall the equations for the horizontal coordinate ($x$) and vertical coordinate ($y$) of a projectile at time $t$, if it is launched from the origin with an initial velocity $v_o$ at an angle $\theta_o$ with the horizontal.

Describe the direction of the velocity vector and the acceleration vector for an object in uniform circular motion. Explain if these vectors are constant.

Explain the head-to-tail graphical method, also known as the triangle method, for adding two vectors $\mathbf{A}$ and $\mathbf{B}$.

Two forces, $\mathbf{F}_1$ of magnitude $10 \text{ N}$ and $\mathbf{F}_2$ of magnitude $8 \text{ N}$, act on a particle. If the angle between the two forces is $60^\circ$, calculate the magnitude of the resultant force.

A ceiling fan completes 1200 revolutions in one minute. Calculate its angular speed in radians per second.

A car moves on a circular track of radius $100 \text{ m}$ at a constant speed of $72 \text{ km/h}$. Calculate its centripetal acceleration.

Propose a method to determine the magnitude of the centripetal acceleration of an athlete running at a constant speed on a circular track, using only a measuring tape and a stopwatch. Detail the steps and the final formula.

Evaluate whether the vectors $\mathbf{A} = 3\hat{\mathbf{i}} + 4\hat{\mathbf{j}}$ and $\mathbf{B} = 4\hat{\mathbf{i}} - 3\hat{\mathbf{j}}$ are perpendicular. Justify your conclusion using the concept of direction or slope.

Formulate the general equation for the path of a projectile, $y(x)$, and justify why its trajectory is a parabola.

A projectile is launched with an initial speed $v_0$ at an angle $\theta_0$ with the horizontal. A second object is thrown vertically upwards with the same initial speed $v_0$. Evaluate which of the two will reach a greater maximum height. Justify your answer with mathematical derivations.

Design three distinct physical scenarios where a particle's velocity vector $\mathbf{v}$ and acceleration vector $\mathbf{a}$ are (a) parallel, (b) anti-parallel, and (c) perpendicular. For each case, justify your choice with a brief explanation and a real-world example.

Create and solve a problem to find a vector $\mathbf{C}$ in the $x-y$ plane, given that the sum of three vectors $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ is a null vector. Let $\mathbf{A}$ be a vector of magnitude $10$ units along the positive $x$-axis, and $\mathbf{B}$ be a vector of magnitude $15$ units at an angle of $120^{\circ}$ counter-clockwise from the positive $x$-axis.

List and explain the commutative and associative laws of vector addition. Also, define a null vector and state one of its properties.

An insect moves in a circular groove of radius $12 \text{ cm}$ and completes 7 revolutions in $100 \text{ s}$. Identify the physical quantity represented by each of these values and recall the formula for angular speed $\omega$.

Explain the concept of resolving a vector in a two-dimensional plane. If a vector $\mathbf{A}$ has magnitude $A$ and makes an angle $\theta$ with the positive x-axis, recall the expressions for its components $A_x$ and $A_y$ and write the vector in its component form.

The position of a particle is described by the vector $\mathbf{r}(t) = (3t^2 \hat{\mathbf{i}} + 5t \hat{\mathbf{j}} - 2t^3 \hat{\mathbf{k}}) \text{ m}$. Solve for the velocity vector $\mathbf{v}(t)$ and the acceleration vector $\mathbf{a}(t)$. Then, calculate the magnitude of the velocity at $t = 2 \text{ s}$.

A stone is thrown horizontally with a speed of $20 \text{ m/s}$ from the top of a cliff $80 \text{ m}$ high. Solve for the horizontal distance from the base of the cliff where the stone strikes the ground. (Neglect air resistance and take $g = 9.8 \text{ m/s}^2$)

Rain is falling vertically downwards with a speed of $30 \text{ m/s}$. A woman rides a bicycle with a speed of $10 \text{ m/s}$ in the east to west direction. Analyze the situation to determine the direction in which she should hold her umbrella to protect herself from the rain.

Design an experiment to justify Galileo's statement that 'for elevations which exceed or fall short of 45° by equal amounts, the ranges are equal'. Propose the necessary equipment, procedure, and method of analysis.

Formulate a problem involving a particle moving in the $x-y$ plane with a constant acceleration. The problem should require finding the trajectory of the particle in the form $y(x)$. Propose a complete solution for the problem you formulated.