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Q3.16(b):
Read each statement below carefully and state, with reasons, if it is true or false : (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point
Solution :
Statement Analysis
The statement is: "The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point."
Verdict: True.
Theoretical Justification
By the definition of instantaneous velocity $\vec{v}$, the velocity is the limit of the average velocity as the time interval $\Delta t$ approaches zero [Per the definition of the derivative in kinematics]:
$$\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$$
Where $\Delta \vec{r}$ is the displacement vector between two points $P$ and $Q$ on the path. As $\Delta t \to 0$, point $Q$ approaches point $P$. The direction of the displacement vector $\Delta \vec{r}$ (the secant line) converges to the direction of the tangent at point $P$.
Geometrical Illustration
Logical Derivation
- Step 1: Define the position vector $\vec{r}(t)$ at time $t$ and $\vec{r}(t + \Delta t)$ at $t + \Delta t$.
- Step 2: The displacement vector $\Delta \vec{r} = \vec{r}(t + \Delta t) - \vec{r}(t)$ represents the chord connecting two points on the trajectory.
- Step 3: As the interval $\Delta t \to 0$, the ratio $\frac{\Delta \vec{r}}{\Delta t}$ approaches the derivative $\frac{d\vec{r}}{dt}$, which geometrically defines the slope of the curve at point $P$.
- Step 4: Because the derivative of the position vector with respect to time represents the instantaneous rate of change, its direction must be collinear with the line segment formed in the limit, which is the tangent line [By the Fundamental Theorem of Calculus in vector kinematics].
Conclusion
The velocity vector is defined as the time derivative of the position vector. Geometrically, the limit of the displacement vector as $\Delta t$ approaches zero is the tangent vector at that point on the trajectory. Thus, the direction of motion is always tangential to the path at any given instant.
Final Solution: The statement is True, because the instantaneous velocity is defined as the derivative of the position vector $d\vec{r}/dt$, which by geometric construction is the tangent to the path at that coordinate.
More Questions from Class 11 Physics Motion in a Plane EXERCISES
- Q3.1: State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
- Q3.10: On an open ground, a motorist follows a track that turns to his left by an angle of (60^{\circ}) after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
- Q3.11(a): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi,
- Q3.11(b): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (b) the magnitude of average velocity ? Are the two equal ?
- Q3.12: The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of (40 m s^{-1}) can go without hitting the ceiling of the hall ?
- Q3.13: A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball ?
- Q3.14: A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone ?
- Q3.15: An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.
- Q3.16(a): Read each statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre
- Q3.16(c): Read each statement below carefully and state, with reasons, if it is true or false : (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector
- Q3.17(a): The position of a particle is given by (r = 3.0t \hat{i} - 2.0t^{2} \hat{j} + 4.0 \hat{k} m) where t is in seconds and the coefficients have the proper units for r to be in metres. (a) Find the (v) and (a) of the particle?
- Q3.17(b): The position of a particle is given by (r = 3.0t \hat{i} - 2.0t^{2} \hat{j} + 4.0 \hat{k} m) where t is in seconds and the coefficients have the proper units for r to be in metres. (b) What is the magnitude and direction of velocity of the particle at (t = 2.0 s) ?
- Q3.18(a): A particle starts from the origin at (t = 0 s) with a velocity of (10.0 \hat{j} m/s) and moves in the x-y plane with a constant acceleration of ((8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}). (a) At what time is the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
- Q3.18(b): A particle starts from the origin at (t = 0 s) with a velocity of (10.0 \hat{j} m/s) and moves in the x-y plane with a constant acceleration of ((8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}). (b) What is the speed of the particle at the time ?
- Q3.19: (\hat{i}) and (\hat{j}) are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors (\hat{i} + \hat{j}), and (\hat{i} - \hat{j})? What are the components of a vector (A= 2 \hat{i} + 3 \hat{j}) along the directions of (\hat{i} + \hat{j}) and (\hat{i} - \hat{j})? [You may use graphical method]
- Q3.2: Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
- Q3.20(a): For any arbitrary motion in space, which of the following relations are true : (a) (v_{average} = (\frac{1}{2}) (v(t_{1}) + v(t_{2})))
- Q3.20(b): For any arbitrary motion in space, which of the following relations are true : (b) (v_{average} = \frac{[r(t_{2}) - r(t_{1})]}{t_{2} – t_{1}})
- Q3.20(c): For any arbitrary motion in space, which of the following relations are true : (c) (v(t) = v(0) + a t)
- Q3.20(d): For any arbitrary motion in space, which of the following relations are true : (d) (r(t) = r(0) + v(0)t + (\frac{1}{2}) a t^{2})
- Q3.20(e): For any arbitrary motion in space, which of the following relations are true : (e) (a_{average} = \frac{[v(t_{2}) - v(t_{1})]}{t_{2} – t_{1}}) (The ‘average’ stands for average of the quantity over the time interval (t_{1}) to (t_{2}))
- Q3.21(a): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process
- Q3.21(b): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (b) can never take negative values
- Q3.21(c): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless
- Q3.21(d): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (d) does not vary from one point to another in space
- Q3.21(e): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (e) has the same value for observers with different orientations of axes.
- Q3.22: An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s a part is (30^{\circ}), what is the speed of the aircraft ?
- Q3.3: Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
- Q3.4(a): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars,
- Q3.4(b): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (b) adding a scalar to a vector of the same dimensions ,
- Q3.4(c): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (c) multiplying any vector by any scalar,
- Q3.4(d): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars,
- Q3.4(e): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors,
- Q3.4(f): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.
- Q3.5(a): Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar,
- Q3.5(b): Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar,
- Q3.5(c): Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.
- Q3.5(d): Read each statement below carefully and state with reasons, if it is true or false : (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
- Q3.5(e): Read each statement below carefully and state with reasons, if it is true or false : (e) Three vectors not lying in a plane can never add up to give a null vector.
- Q3.6(a): Establish the following vector inequalities geometrically or otherwise : (a) (|a+b| < |a| + |b|) When does the equality sign above apply?
- Q3.6(b): Establish the following vector inequalities geometrically or otherwise : (b) (|a+b| > ||a| −|b||) When does the equality sign above apply?
- Q3.6(c): Establish the following vector inequalities geometrically or otherwise : (c) (|a−b| < |a| + |b|) When does the equality sign above apply?
- Q3.6(d): Establish the following vector inequalities geometrically or otherwise : (d) (|a−b| > ||a| − |b||) When does the equality sign above apply?
- Q3.7(a): Given (a + b + c + d = 0), which of the following statements are correct : (a) (a), (b), (c), and (d) must each be a null vector,
- Q3.7(b): Given (a + b + c + d = 0), which of the following statements are correct : (b) The magnitude of ((a + c)) equals the magnitude of ((b + d)),
- Q3.7(c): Given (a + b + c + d = 0), which of the following statements are correct : (c) The magnitude of (a) can never be greater than the sum of the magnitudes of (b), (c), and (d),
- Q3.7(d): Given (a + b + c + d = 0), which of the following statements are correct : (d) (b + c) must lie in the plane of (a) and (d) if (a) and (d) are not collinear, and in the line of (a) and (d), if they are collinear ?
- Q3.8: Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 3.19. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?
- Q3.9(a): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (a) net displacement,
- Q3.9(b): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (b) average velocity,
- Q3.9(c): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?
- Q4.1: State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
- Q4.10: On an open ground, a motorist follows a track that turns to his left by an angle of $60^{\circ}$ after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
- Q4.11(a): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi,
- Q4.11(b): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (b) the magnitude of average velocity ? Are the two equal ?
- Q4.12: Rain is falling vertically with a speed of $30 m s^{-1}$. A woman rides a bicycle with a speed of $10 m s^{-1}$ in the north to south direction. What is the direction in which she should hold her umbrella ?
- Q4.13: A man can swim with a speed of $4.0 km/h$ in still water. How long does he take to cross a river $1.0 km$ wide if the river flows steadily at $3.0 km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank ?
- Q4.14: In a harbour, wind is blowing at the speed of $72 km/h$ and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of $51 km/h$ to the north, what is the direction of the flag on the mast of the boat ?
- Q4.15: The ceiling of a long hall is $25 m$ high. What is the maximum horizontal distance that a ball thrown with a speed of $40 m s^{-1}$ can go without hitting the ceiling of the hall ?
- Q4.16: A cricketer can throw a ball to a maximum horizontal distance of $100 m$. How much high above the ground can the cricketer throw the same ball ?
- Q4.17: A stone tied to the end of a string $80 cm$ long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in $25 s$, what is the magnitude and direction of acceleration of the stone ?
- Q4.18: An aircraft executes a horizontal loop of radius $1.00 km$ with a steady speed of $900 km/h$. Compare its centripetal acceleration with the acceleration due to gravity.
- Q4.19(a): Read each statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre
- Q4.19(b): Read each statement below carefully and state, with reasons, if it is true or false : (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point
- Q4.19(c): Read each statement below carefully and state, with reasons, if it is true or false : (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector
- Q4.2: Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
- Q4.20(a): The position of a particle is given by $\boldsymbol{r} = 3.0t \hat{i} - 2.0t^2 \hat{j} + 4.0 \hat{k} m$ where $t$ is in seconds and the coefficients have the proper units for $\boldsymbol{r}$ to be in metres. (a) Find the $\boldsymbol{v}$ and $\boldsymbol{a}$ of the particle?
- Q4.20(b): The position of a particle is given by $\boldsymbol{r} = 3.0t \hat{i} - 2.0t^2 \hat{j} + 4.0 \hat{k} m$ where $t$ is in seconds and the coefficients have the proper units for $\boldsymbol{r}$ to be in metres. (b) What is the magnitude and direction of velocity of the particle at $t = 2.0 s$ ?
- Q4.21(a): A particle starts from the origin at $t = 0 s$ with a velocity of $10.0 \hat{j} m/s$ and moves in the x-y plane with a constant acceleration of $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (a) At what time is the x- coordinate of the particle $16 m$? What is the y-coordinate of the particle at that time?
- Q4.21(b): A particle starts from the origin at $t = 0 s$ with a velocity of $10.0 \hat{j} m/s$ and moves in the x-y plane with a constant acceleration of $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (b) What is the speed of the particle at the time ?
- Q4.22: $\hat{i}$ and $\hat{j}$ are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors $\hat{i} + \hat{j}$ , and $\hat{i} - \hat{j}$ ? What are the components of a vector $A= 2 \hat{i} + 3 \hat{j}$ along the directions of $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$ ? [You may use graphical method]
- Q4.23(a): For any arbitrary motion in space, which of the following relations are true : (a) $v_{average} = (1/2) (v (t_1) + v (t_2))$
- Q4.23(b): For any arbitrary motion in space, which of the following relations are true : (b) $v_{average} = [r(t_2) - r(t_1) ] /(t_2 - t_1)$
- Q4.23(c): For any arbitrary motion in space, which of the following relations are true : (c) $v (t) = v (0) + a t$
- Q4.23(d): For any arbitrary motion in space, which of the following relations are true : (d) $r (t) = r (0) + v (0) t + (1/2) a t^2$
- Q4.23(e): For any arbitrary motion in space, which of the following relations are true : (e) $a_{average} =[ v (t_2) - v (t_1 )] /( t_2 - t_1)$ (The ‘average’ stands for average of the quantity over the time interval $t_1$ to $t_2$)
- Q4.24(a): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process
- Q4.24(b): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (b) can never take negative values
- Q4.24(c): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless
- Q4.24(d): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (d) does not vary from one point to another in space
- Q4.24(e): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (e) has the same value for observers with different orientations of axes.
- Q4.25: An aircraft is flying at a height of $3400 m$ above the ground. If the angle subtended at a ground observation point by the aircraft positions $10.0 s$ apart is $30^{\circ}$, what is the speed of the aircraft ?
- Q4.3: Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
- Q4.4(a): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars,
- Q4.4(b): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (b) adding a scalar to a vector of the same dimensions ,
- Q4.4(c): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (c) multiplying any vector by any scalar,
- Q4.4(d): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars,
- Q4.4(e): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors,
- Q4.4(f): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.
- Q4.5(a): Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar,
- Q4.5(b): Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar,
- Q4.5(c): Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.
- Q4.5(d): Read each statement below carefully and state with reasons, if it is true or false : (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
- Q4.5(e): Read each statement below carefully and state with reasons, if it is true or false : (e) Three vectors not lying in a plane can never add up to give a null vector.
- Q4.6(a): Establish the following vector inequalities geometrically or otherwise : (a) $|a+b| < |a| + |b|$. When does the equality sign above apply?
- Q4.6(b): Establish the following vector inequalities geometrically or otherwise : (b) $|a+b| > ||a| -|b||$. When does the equality sign above apply?
- Q4.6(c): Establish the following vector inequalities geometrically or otherwise : (c) $|a-b| < |a| + |b|$. When does the equality sign above apply?
- Q4.6(d): Establish the following vector inequalities geometrically or otherwise : (d) $|a-b| > ||a| - |b||$. When does the equality sign above apply?
- Q4.7(a): Given $a + b + c + d = 0$, which of the following statements are correct : (a) $a$, $b$, $c$, and $d$ must each be a null vector,
- Q4.7(b): Given $a + b + c + d = 0$, which of the following statements are correct : (b) The magnitude of $(a + c)$ equals the magnitude of $( b + d)$,
- Q4.7(c): Given $a + b + c + d = 0$, which of the following statements are correct : (c) The magnitude of $a$ can never be greater than the sum of the magnitudes of $b$, $c$, and $d$,
- Q4.7(d): Given $a + b + c + d = 0$, which of the following statements are correct : (d) $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of $a$ and $d$, if they are collinear ?
- Q4.8: Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?
- Q4.9(a): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement,
- Q4.9(b): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (b) average velocity,
- Q4.9(c): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?
CBSE Solutions for Class 11 Physics Motion in a Plane
Chapters in CBSE - Class 11 Physics
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NEET-UG Coaching, Class 9 Tuition
and more
+4
(40)
Top Tutor
Pandeypur, Varanasi
11 yrs of
Exp
7 students
2131 hrs of classes completed
600
per hour
10 years' experience of teaching
Online Classes
I am Mathematics teacher, passionate to teach Mathematics of different states of India. I am teaching class 11th since 2011 only in a reputed CBSE BOARD school. Several students who are taught by me are studying in different IITs.
Bobby P
Excellent Maths teacher, sir taught 9,10,11,12 cbse, My son improved a lot. Even gives test, Top board tutor.
Skills:
Mathematics
,
Applied Mathematics
Also teaches:
Class 10 Tuition, Vedic Maths Training
and more
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