Motion in a Plane

### A man wants to reach from A to the opposite comer of the square C. The sides of the square are 100 m. A central square of 50 m × 50 m is filled with sand. Outside this square, he can walk at a speed 1 m/s. In the central square, he walk only at a speed of v m/s. What is smallest value of v for which he can reach faster via a straight path through the sand than any path in the square outside the sand?

Answer: As depicted in the diagram, APQC represents the path taken by the guy through the sand, the time it took him to get from A to C, and the distance travelled by him.  \begin{aligned}...

### A river is flowing due east with a speed 3 m/s. A swimmer can swim in still water at a speed of 4 m/s.c) from two different cases as mentioned in a) and b) above, in which case will he reach opposite bank in shorter time?

Answer: c) In case (a), Time taken by the swimmer to cross the river, $t_{1}=\frac{d}{v_{s}}=\frac{d}{4} s$ In case (b), Time taken by the swimmer to cross the river,...

### A river is flowing due east with a speed 3 m/s. A swimmer can swim in still water at a speed of 4 m/s. a) if swimmer starts swimming due north, what will be his resultant velocity? b) if he wants to start from point A on south bank and reach opposite point B on north bank, i) which direction should he swim? ii) what will be his resultant speed?

Answer: Given, The river's velocity, vr, is 3 meters per second. vs = 4 m/s is the speed of the swimmer. a) When the swimmer swims due north, the y-component will have a velocity of 4 meters per...

### A girl riding a bicycle with a speed of 5 m/s towards north direction, observes rain falling vertically down. If she increases her speed to 10 m/s, rain appears to meet her at 45o to the vertical. What is the speed of the rain? In what direction does rain fall as observed by a ground based observer?

Suppose that Vrg is the velocity of the rain drop that appears to the female observer. All of the vectors are drawn with reference to the frame from the ground up, save for one. Let's say the rain...

### A particle falling vertically from a height hits a plane surface inclined to horizontal at an angle θ with speed v0 and rebounds elastically. Find the distance along the plane where it will hit second time.

Answer: When the x and y axes are set in the manner depicted in the diagram, the missile moves from point O to point A. y = 0 uy = vo cos θ ay = -g cos θ t = T Using the kinematics for the y-axis y...

### A particle is projected in air at an angle β to a surface which itself is inclined at an angle α to the horizontal. a) find an expression of range on the plane surface b) time of flight c) β at which range will be maximum

Answer: This problem has two solutions: i)The point P where the particle impacts the plane is the parabola/straight line intersection.  In other words, it's not a straight line. ii) It...

### A gun can fire shells with maximum speed and the maximum horizontal range that can be achieved is . If a target farther away by distance has to be hit with the same gun, show that it could be achieved by raising the gun to a height at least

Answer: $R=\frac{v_{0}^{2}}{g}$ is the maximum range. As a result, the projection angle is 45 degrees. The gun is raised to a height of h in order to hit the target. Negative is used to the vertical...

### A hill is 500 m high. Supplies are to be sent across the hill using a canon that can hurl packets at a speed of 125 m/s over the hill. The canon is located at a distance of 800 m from the foot of hill and can be moved on the ground at a speed of 2 m/s so that its distance from the hill can be adjusted. What is the shortest time in which a packet can reach on the ground across the hill? Take g = 10 m/s2.

Answer: The speed of packets is 125 m/s, the height of the hill is 500 m, and the distance between the cannon and the foot of the hill is 800 m, according to the problem. Ideally, the vertical...

### If and , then match the relations in column with the angle between and in column .

Column I Column II a) |A×B|=0 i) θ = 30o b) |A×B|=8 ii) θ = 45o c) |A×B|=4 iii) θ = 90o d) |A×B|=4√2 iv) θ = 0o Answer: a) matches with iv) b) matches with iii) c) matches with i) d) matches with...

### If and , then match the relations in column I with the angle between between and in column II.

Column I Column II a) A.B = 0 i) θ = 0 b) A.B = +8 ii) θ = 90o c) A.B = 4 iii) θ = 180o d) A.B = -8 iv) θ = 0o   Answer: a) matches with ii) b) matches with i) c) matches with iv) d) matches...

### Given below in column I are the relations between vectors a, b, and c and in column II are the orientations of a, b, and c in the XY plane. Match the relation in column I to correct orientations in column II.

Answer: a) matches with iv) b) matches with iii) c) matches with i) d) matches with ii)

### a) Earth can be thought of as a sphere of radius 6400 km. Any object is performing circular motion around the axis of earth due to earth’s rotation. What is acceleration of object on the surface of the earth towards its centre? What is it at latitude θ? How does these accelerations compare with g = 9.8 m/s2? b) Earth also moves in circular orbit around sun once every year with an orbital radius of 1.5 × 1011m. What is the acceleration of earth towards the centre of the sun? How does this acceleration compare with g = 9.8 m/s2?

Answer: (a) According to the question, we have been given that, Radius of the earth $(R)=6400 km =6.4 \times 10^{6} m$. Time period of the motion $(T)=1$ day $=24 \times 60 \times 60 s =86400 s$ As...

### A fighter plane is flying horizontally at an altitude of 1.5 km with speed 720 km/h. At what angle of sight when the target is seen, should the pilot drop the bomb in order to attack the target?

Answer: Given, u = 720 km/h = 200 m/s If t is the moment at which the pilot detonates the bomb, then Q is the point vertically above the target T at which the bomb is dropped. The bomb's horizontal...

### In dealing with motion of projectile in air, we ignore effect of air resistance on motion. This gives trajectory as a parabola as you have studied. What would the trajectory look like if air resistance is included? Sketch such a trajectory and explain why you have drawn it that way.

The vertical and horizontal velocity of a projectile reduces due to air resistance. The formula for lowering the height of motion is as follows: R = (u2/g) sin 2θ Hmax = u2 sin2 θ/2g The graphic...

### A boy throws a ball in air at 60o to the horizontal along a road with a speed of 10 m/s. Another boy sitting in a passing by car observes the ball. Sketch the motion of the ball as observed by the bot in the car, if car has a speed of 18 km/h. Give explanation to support your diagram.

Answer: Given, u = 36 km/h = 10 m/s ux = u cos 60o = 5 m/s Speed of the car in the direction of motion of ball = (18)(5/18) = 5 m/s The boy throws a ball when a car goes by. ...

### A boy travelling in an open car moving on a labelled road with constant speed tosses a ball vertically up in the air and catches it back. Sketch the motion of the ball as observed by a boy standing on the footpath. Give explanation to support your diagram.

Answer: Given, v denotes the vertical velocity of the ball that the boy is holding. In this equation, u = horizontal velocity of the ball multiplied by the velocity of the car. The diagram shown...

### A, B, and C are three non-collinear, non co-planar vectors. What can you say about direction of A(B×C)?

Answer: When vector B and C are added together, the direction of the product is perpendicular to the plane that contains vector B and C, which is based on the Right hand grip rule. In this case, the...

### A football is kicked into the air vertically upwards. What is its a) acceleration b) velocity at the highest point

Answer: a) When a football is kicked vertically upwards into the air, as depicted in the diagram below, the situation is considered to be "upwards." It is called acceleration owing to gravity when...

### A ball is thrown from a roof top at an angle of 45o above the horizontal. It hits the ground a few seconds later. At what point during its motion, does the ball have a) greatest speed b) smallest speed c) greatest acceleration? Explain

Answer: a) Assume the ball is thrown $45^o$ from O. The ball's height increases from O to A, lowering its KE. The speed increases as the height lowers from A to B, until it reaches the initial speed...

### A particle is projected in air at some angle to the horizontal, moves along parabola as shown in the figure, where x and y indicate horizontal and vertical directions respectively. Show in the diagram, direction of velocity and acceleration at points A, B, and C.

Answer: The projectile motion is parabolic. The velocity is always tangential to A, B, and C. The trajectory reaches its greatest height at B. So Bvy = 0 and u cos. We know that acceleration follows...

### A cyclist starts from centre O of a circular park of radius 1 km and moves along the path OPRQO as shown in the figure. If he maintains constant speed of 10 m/s, what is his acceleration at point R in magnitude and direction?

Answer: The circular path has a radius of 1 km, with O as the center and R as the distance between the two points. The biker is moving at a speed of 10 meters per second....

### For two vectors A and B, is always true when a)     b)     c) and and are parallel or antiparallel d) when either or is zero

Answer: The correct option is a) & d) Given in question $|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|$, it can be assertive when $| A |=0$ or $| B |=0$ or both are zero. The provided statement can also...

### For a particle performing uniform circular motion, choose the correct statement from the following: a) magnitude of particle velocity (speed) remains constant b) particle velocity remains directed perpendicular to radius vector c) direction of acceleration keeps changing as particle moves d) angular momentum is constant in magnitude but direction keep changing

Answer: The correct answer is a) magnitude of particle velocity (speed) remains constant, b) particle velocity remains directed perpendicular to radius vector and c) direction of acceleration keeps...

### Following are four different relations about displacement, velocity, and acceleration for the motion of a particle in general. Choose the incorrect one (s): a) = 1/2 [v(t1) + v(t2)] b) = r(t2)-r(t1)/t2-t2 c) r = 1/2 [v(t2)-v(t1)](t2-t1) d) = v(t2)-v(t1)/t2-t1

Answer: \text { The correct answer is a) } v_{ av }=1 / 2\left[ v \left( t _{1}\right)+ v \left( t _{2}\right)\right] \text { and c) } r =1 / 2\left[ v \left( t _{2}\right)- v \left( t...

### A particle slides down a frictionless parabolic track starting from rest at point A. Point B is at the vertex of parabola and point C is at a height less than that of point A. After C, the particle moves freely in air as a projectile. If the particle reaches highest point at P, then a) KE at P = KE at B b) height at P = height at A c) total energy at P = total energy at A d) time of travel from A to B = time of travel from B to P

Answer: The correct answer is c) total energy at P = total energy at A Because energy is always conserved (unless in inelastic collisions), the total energy at A and P will always be equal....

### Two particles are projected in air with speed , at angles θ1 and θ2 to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then tick the right choices a) angle of project: q1 > q2 b) time of flight: T1 > T2 c) horizontal range: R1 > R2 d) total energy: U1 > U2

Answer: The correct answer is a) angle of the project: q1 > q2 and b) time of flight: T1 > T2 Assuming this is true, two particles are pushed into the air at a speed of u and at angles of 1...

### It is found that . This necessarily implies a) b) A, B are antiparallel c) A, B are perpendicular d) A.B

Answer: The correct answer is a) B = 0 Its mean vector B will have 0 magnitudes, and there will be no direction in which it moves. A→+B→=A→\overrightarrow{ A }+\overrightarrow{...

### Three vectors A, B, and C add up to zero. Find which is false, a) vector is not zero unless vectors are parallel d) vector is not zero unless vectors are parallel c) if vectors define a plane, is in that plane d) such that

Answer: The correct answer is $c$ ) if vectors $A, B, C$ define a plane, $(A \times B) C$ is in that plane and d) $(A \times B) \cdot C=$ $|A \| B||C| \quad$ such that $C ^{2}= A ^{2}+ B ^{2}$ Given...

### In a two dimensional motion, instantaneous speed is a positive constant. Then which of the following are necessarily true? a) the acceleration of the particle is zero b) the acceleration of the particle is bounded c) the acceleration of the particle is necessarily in the plane of motion d) the particle must be undergoing a uniform circular motion

Answer: The correct answer is d) the particle must be undergoing a uniform circular motion In two dimensions, instantaneous speed $v_0$ is positive. The particle's acceleration must be in the plane...

### In a two dimensional motion, instantaneous speed is a positive constant. Then which of the following are necessarily true? a) the average velocity is not zero at any time b) average acceleration must always vanish c) displacements in equal time intervals are equal d) equal path lengths are traversed in equal intervals

Answer: The correct answer is d) equal path lengths are traversed in equal intervals Given that the immediate speed $v_0$ is a positive constant, this motion is two-dimensional. Because acceleration...

### Consider the quantities pressure, power, energy, impulse gravitational potential, electric charge, temperature, area. Out of these, the only vector quantities are a) impulse, pressure, and area b) impulse and area c) area and gravitational potential d) impulse and pressure

Answer: The correct answer is b) impulse and area We know that impulse is defined as $J = F \cdot \Delta t =\Delta p$, where $F$ is force, At is time length, and $\Delta p$ represents momentum...

### 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.

Answer : (c) True The direction of the acceleration vector in a uniform circular motion (UCM) is toward the circle's centre. It does, however, vary over time. A null vector is the average of these...

### Read each statement below carefully and state, with reasons, if it is true or false: (a) The net acceleration of a particle in a circular motion is always along the radius of the circle towards the centre. (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point.

Answer : (a) False Only in the situation of uniform circular motion is the net acceleration of a particle in a circular motion directed along the radius of the circle toward the center. (b) True...

### 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 direction and magnitude of the acceleration of the stone?

Answer - According to the question, the length of the string, l is 80 cm, or 0.8 m and the number of revolutions is 14 And the time taken = 25 s Expression for the frequency is given as follows -...

### 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?

answer - We are given that the maximum horizontal distance is R = 100 m When the angle of projection is 45 degrees, only then the cricketer can toss the ball the maximum horizontal distance....

### The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball is thrown with a speed of 40 ms^{-1}ms −1 can go without hitting the ceiling of the hall?

Answer : According to the question, the speed of the ball is u = 40 ms-1 And the maximum height is h = 25 m The maximum height obtained by a body projected at an angle θ in projectile motion is...

### In a harbour, the 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?

Answer - According to the question, the velocity of the boat is 51 km/h and the velocity of the wind is 72 km/h. The flag is flapping in the direction of northeast. It indicates that the wind is...

### A man can swim with a speed of 4 km/h in still water. How long does he take to cross a river 1 km wide if the river flows steadily at 3 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?

Answer : According to the question, speed of the man is ${{v}_{m}}=4km/h$ and the width of the river is 1 km Then the time taken in crossing the river can be determined as follows :...

### 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}ms −1 in the north to south direction. What is the direction in which she should hold her umbrella?

Answer : Here, vc denotes theVelocity of the cyclist = 10 m/s vr denotes the Velocity of falling rain = 30 m/s The woman must hold her umbrella in the direction of the relative velocity (v) of the...

### A passenger arriving in a new town wants 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. (a) What is the average speed of the taxi? (b) What is the magnitude of average velocity? Are the two equal?

Answer : (a) According to the question, the total distance travelled is 23 km and the total time taken is 28 minutes. Time Taken (in hours) = 28/60 h Average speed is given as follows : Average...

### On an open ground, a motorist follows a track that turns to his left by an angle of 600 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

Answer - As shown in the diagram, the motorist's path is a regular hexagon with a 500-meter side. Let us suppose that the motorist starts from point P and then he takes the third turn at S....

### 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 (i) Net displacement (ii) Average velocity and (iii) The average speed of the cyclist.

Answer - (i) Displacement refers to the distance between the body's original and ultimate positions. In 20 minutes, the cyclist returns to the point where he began. As a result, there is no...

### 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?

Answer - The smallest distance between a particle's initial and final coordinates determines displacement. In the example, all of the girls begin at point P and work their way to point Q. Their...

### Given that l + m + n + o = 0, which of the given statements are true: (c) The magnitude of l can never be greater than the sum of the magnitudes of m, n and o. (d) m + n must lie in the plane of l and o if l and o are not collinear, and in the line of l and o, if they are collinear?

Answer - (c) True We can write the given equation as => l = (m + n + o) Taking mode on both the sides, we get - | l | = | m + n + o | or | l | <= | m + n + o | The magnitude of l is equal to...

### Given that l + m + n + o = 0, which of the given statements are true: (a) l, m, n and o each must be a null vector. (b) The magnitude of (l + n) equals the magnitude of (m+ o).

Answer - (a) False It is not necessary for all four supplied vectors to be null vectors in order for l + m + n + o = 0. There are more combinations that can result in a zero sum. (b) True l + m + n...

### Establish the following vector inequalities geometrically or otherwise:

$\left| \vec{a}-\vec{b} \right|\ge \left| \left| {\vec{a}} \right|-\left| {\vec{b}} \right| \right|$ Answer - Let two vectors $\vec{a}$and $\vec{b}$represent two adjacent sides of a parallelogram...

### Establish the following vector inequalities geometrically or otherwise:

$\left| \vec{a}-\vec{b} \right|\le \left| {\vec{a}} \right|+\left| {\vec{b}} \right|$ Answer - Let two vectors $\vec{a}$and $\vec{b}$represent two adjacent sides of a parallelogram PQRS, as show...

### Establish the following vector inequalities geometrically or otherwise:

$\left| \vec{a}+\vec{b} \right|\ge \left| \left| {\vec{a}} \right|-\left| {\vec{b}} \right| \right|$ Answer : Let two vectors $\vec{a}$and $\vec{b}$represent two adjacent sides of a parallelogram...

### Establish the following vector inequalities geometrically or otherwise:

$\left| \overrightarrow{a}+\overrightarrow{b} \right|\le \left| \overrightarrow{a} \right|+\left| {\vec{b}} \right|$ Answer - Let two vectors $\vec{a}$and $\vec{b}$represent two adjacent sides of...

### 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.

Answer : (e) True : The three vectors are not in a plane, it is not possible to represent the sides of a triangle in the same order.

### 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 (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 the average velocity of the particle over the same interval of time

Answer : (c) False: The total length of the path is a scalar quantity, and the displacement is a vector quantity. Therefore, the total length of the path is always greater than the amplitude of the...

### 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 (b) Each component of a vector is always a scalar

Answer : a) True: The magnitude of the vector is a number. Hence it is a scalar quantity. (b) False: Every component of a vector is also a vector.

### State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) Addition of any two scalars (b) Adding a scalar to a vector which has the same dimensions (c) Multiplying a vector by any scalar (d) Multiplying any two scalars (e) Adding any two vectors (f) Addition of a vector component to the same vector.

Answer : (a) Meaningful: Adding two scalar quantities makes sense only if they both represent the same physical quantity. (b) Not Meaningful: The addition of a vector quantity to a scalar quantity...

### From the following identify the vector quantities : Pressure, temperature, energy, time, gravitational potential, power, total path length, charge, coefficient of friction, impulse.

Answer : Impulse is the product of force and time. Since force is a vector, its time product is a scalar, giving a vector.