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**Q.51 A projectile is fired at an angle of 15 ^{∘} to the horizontal with the speed υ. If another projectile is projected with the same speed, then at what angle with the horizontal it must be projected so as to have the same range?**

**Answer:** For same R and υ, the sum of the two angles of projection is 90^{∘}. As one angle is of 15^{∘}, other should be 90^{∘}−15^{∘} =75^{∘}.**Q.52 Is the maximum height attained by projectile is largest when its horizontal range is maximum?**

**Answer:** No. Horizontal range is maximum when θ=45^{∘} and maximum height attained by projectile is largest when θ=90^{∘}.**Q.53 What will be the effect on maximum height of a projectile when its angle of projection is changed from 30 ^{∘} to 60^{∘}, keeping the same initial velocity of projection?**

**Answer:** Maximum height,

Thus the maximum height becomes three times the original maximum height.**Q.54 A body is projected with speed u at an angle θ to the horizontal to have maximum range. What is the velocity at the highest point?**

**Answer: **For maximum horizontal range, θ=45^{∘} Velocity at highest point = Horizontal component of velocity =u cos 45^{∘}=u/√2**Q.55 What is the angle of projection for a projectile motion whose range R is n times the maximum height H?**

**Answer:** Here R=nH or

ortanθ = 4/n ⇒ θ=tan^{−1}(4/n)

**Q.56 The greatest height to which a man can throw a stone is h. What will be the greatest distance up to which he can throw the stone?****Answer:** Maximum height attained, Clearly, H will be maximum when sinθ=1 ∴ H_{max}= u^{2}/g =2h.

**Q.57 A particle is projected at an angle θ from the horizontal with kinetic energy K. What is the kinetic energy of the particle at the highest point?**

**Answer:** If the particle is projected with velocity υ, then its initial kinetic energy is K=1/2 mυ^{2} Velocity of the projectile at the highest point =υcosθ ∴K.E. of the particle at the highest point = 1/2 m(υcosθ)^{2}=(1/2 mυ^{2})cos^{2}θ=Kcos^{2}θ.**Q.58 A man can jump on the moon six times as high as on the earth. Why?**

**Answer:** Maximum height attained on earth, Acceleration due to gravity on moon, g′=g/6 ∴ Maximum height attained on moon,

So one can jump on moon six times as high as on earth.**Q.59 A projectile of mass m is fired with velocity υ at an angle θ with the horizontal. What is the change in momentum as it rises to the highest point of the trajectory?**

**Answer:** Horizontal component of velocity remains constant. The vertical component is υ sin θ at the point of projection and zero at the highest point. Magnitude of change in momentum =m(υsinθ−0)=mυsinθ (in the vertical direction)**Q.60 A projectile of mass m is thrown with velocity υ from the ground at an angle of 45 ^{∘} with the horizontal. What is the magnitude of change in momentum between leaving and arriving back at the ground?**

**Answer:** The vertical component of velocity at the point of projection is υ sin θ and at the point of return is −υ sin θ . ∴ Magnitude of change in momentum =m(υsinθ+υsinθ)= 2mυ sinθ = 2mυ sin45^{∘}=√2mυ (in the vertical direction)

**Q.61 A block slides down a smooth inclined plane when released from the top while another falls freely from the same point. Which one of them will strike the ground earlier?**

**Answer:** Acceleration of the freely falling block is g while that of the block sliding down the smooth inclined plane is gsinθ. Asg>gsinθ, so the block falling freely will reach the ground earlier.**Q.62 Why does a tennis ball bounce higher on hills than in plains?**

**Answer:** Maximum height attained by a projectile ∝ 1/g As the value of g is less on hills than on plains, so a tennis ball bounces higher on hills than on plains.**Q.63 A stone is thrown horizontally with a speed from the top of a wall of height h. It strikes the level ground through the foot of the wall at a distance x from the wall. What is the value of x?**

**Answer: **

** ** or

**Q.64 The velocity of a particle is constant in magnitude but not in direction. What is the nature of trajectory?**

**Answer:** The particle moves on a curved path. The instantaneous direction will be tangential to the path at every point.

**Q.65 What is the angle between velocity vector and acceleration vector in uniform circular motion?Answer: **90

**Q.66 For uniform circular motion, does the direction of centripetal acceleration depends upon the sense of rotation?**

**Answer:** The direction of centripetal acceleration does not depend on the clockwise or anticlockwise sense of rotation of the body. It always acts along the radius towards the center of the circle.**Q.67 If both the speed and radius of the circular path of a body are doubled, how will the centripetal acceleration change?**

**Answer:** Original centripetal acceleration, a = v^{2}/r New centripetal acceleration,

**Q.68 A stone tied to the end of a string is whirled in a circle. If the string breaks, the stone flies away tangentially. Why?**

**Answer:** The instantaneous velocity of the stone going around the circular path is always along tangent to the circle. When the string breaks, the centripetal force ceases. Due to inertia, the stone continues its motion along tangent to the circular path.**Q.69 When a knife is sharpened with the help of a rotating grinding stone, the spark always travels tangentially to it. Why?**

**Answer:** When a knife is sharpened, the red hot particles of the grinding stone get separated as sparks. As these particles are in circular motion, hence their velocities are directed tangentially to the grinding stone.**Q.70 Which of the following quantities are independent of the choice of orientation of the coordinate axes : angle between **** is a scalar?Answer:** A vector, its magnitude and the angle between two vectors do not depend on the choice of the orientation of the coordinate axes, so angle between are independent of the orientation of the coordinate axes. But the quantity depends upon the magnitude of the components along x- and y-axes, so it will change with the change in coordinate axes.

**Answer:** Yes, because is represented by the diagonal of the parallelogram drawn with as adjacent sides. The diagonal passes through the common tail of . However, is represented by the other diagonal of the same parallelogram not passing through the common tail of Thus both and lie in plane of the same parallelogram.

**Q.72 Can we apply the commutative and associative laws to vector subtraction also?**

**Answer:** (i) No, we cannot apply commutative law to vector subtraction because (ii) Yes, association law can be applied to vector subtraction because

**Q.73 Can three vectors not in one plane give a zero resultant? Can four vectors do so?**

**Answer:** No, three vectors not in one plane cannot give a zero resultant because the resultant of two vectors will not lie in the plane of the third vector and hence cannot cancel its effect. The resultant of four vectors not in one plane may be a zero vector.**Q.74 What is the difference between the following two data : (i) 8 (5 km/hr, east) (ii) (8 hr) (5 km/hr, east)?**

**Answer:** (i) It is the product of a pure number and a vector (velocity), hence the unit of product is the same as that of vector i.e., the product is a velocity of 40 km/hr, towards east. (ii) It is the product of a scalar (time) and a vector (velocity). Hence the unit of the product will be hr × (km/hr). Thus the product is a displacement of magnitude 40 km, towards east.

**Q.75 Under what condition does the equality: hold good?**

**Answer: ** or or

This implies that are perpendicular to each other.

**Q.76 The sum and difference of two vectors are perpendicular to each other. Prove that the vectors are equal in magnitude.Answer: **As the vectors

.**Q.77 Find the conditions for two vectors to be (i) parallel and (ii) perpendicular to each other.Answer:** and Hence if two vectors are parallel, then their cross product must be zero.

(ii) If and Hence if two vectors are perpendicular, then their dot product must be zero.**Q.78 The magnitude of vector is 2.5 meter and is directed towards east. What will be the magnitudes and directions of the following : **

**Answer: **(a) 2.5 m due west. (b) 1.25 due east. (c) 6.25 m due west. (d) 10 m due east.**Q.79 A person sitting in a moving train throws a ball vertically upwards. How will the ball appear to move to an observer (i) sitting inside the train (ii) standing outside the train? Give reason.**

**Answer:** (i) To the observer inside the train, the ball will appear to move straight vertically upwards and then downwards because with respect to an observer sitting inside the train, the ball has only one velocity acting vertically downward. (ii) To the observer outside the train, the ball will appear to move along the parabolic path because with respect to the observer outside the train, the ball has both horizontal and vertical components of velocity.**Q.80 A bob hung from the ceiling of a room by a string is performing simple harmonic oscillations. What will be the trajectory of the bob, if the string is cut, when bob is (i) at one of its extreme positions, (ii) at its mean position?Answer:** (i) When the string is cut in one of the extreme positions of the bob, the bob will fall vertically downwards. The reason is that when the bob reaches its extreme positions, it comes to rest for a moment. If the string is cut, it will be simply the case of a body falling freely under gravity. (ii) When the string is cut in mean position of the bob, then the bob will move on parabolic path. The reason is that when the bob is at its mean position, it has a velocity parallel to the horizontal. If the string is cut, it will be simply the case of a projectile fired parallel to the horizontal.

**Answer:** (i) When the angle of projection, θ=45^{∘}+α,, the horizontal range is

(ii) When the angle of projection, θ=45^{∘}−α,, the horizontal range is Clearly, R′=R.**Q.82 Two bombs of 5 kg and 10 kg are thrown from a cannon with the same velocity in the same direction, (i) Which bomb will reach the ground first? (ii) If the bombs are thrown in the same direction with different velocity, what would be the effect?Answer: **(i) Both the bombs will reach the ground simultaneously because the time of flight does not depend upon mass of the projectile. (ii) On throwing with different velocities, the bomb thrown with lesser velocity will reach the ground earlier.

**Q.83 Two bodies are thrown with the same initial velocity at angles α and (90 ^{∘} − α) with the horizontal. What will be the ratio of (i) maximum heights attained by them and (ii) their horizontal ranges?Answer: **(i) When the angle of projection α, the maximum height is H

(ii) When the angle of projection is a, the horizontal range is When the angle of projection is(90∘−α), the horizontal range is

∴ R_{1}:R_{2}=1:1**Q.84 At which points on the projectile trajectory is the (i) potential energy maximum (ii) kinetic energy minimum and (iii) total energy maximum?****Answer: **(i) P.E. of a projectile is maximum at its highest point because of its maximum height. It is given by (P.E.)_{H}=mgH=mg.

(ii) K.E. of a projectile is minimum (not zero) at its highest point because of its minimum velocity.

(ii) Total energy at highest point =(P.E.)_{H}+(K.E)_{H} =

Total energy at the point of projection Hence total energy of a projectile is conserved at all points of its motion. The K.E. is maximum at the point of projection.**Q.85 A body A of mass m is thrown with velocity υ at an angle 30 ^{∘} to the horizontal and another body B of the same mass is thrown with velocity υ at an angle of 60^{∘ }to the horizontal, find the ratio of the horizontal range and maximum height of A and B.**

Maximum height,

Hence R_{A}:R_{B}=1:1 and H_{A}:H_{B}=1:3 **Q.86 A ball of mass m is thrown vertically up. Another ball of mass 2m is thrown at an angle θ with the vertical. Both of them remain in air for the same period of time. What is the ratio of the heights attained by the two balls?**

**Answer:** Time of flight of the ball thrown vertically upwards, Time of flight of the ball thrown at an angle θ with the vertical , T_{2}= u_{1}=u_{2}cosθ Now and

**Q.87 Is it important in the long jump that how much height you take for jumping? What factors determine the span of a jump?**

**Answer:** Yes, for the longest jump the player should throw himself at an angle of 45∘ with respect to horizontal. The vertical height for this angle is H=u^{2}sin^{2}45^{∘}/2g=u^{2}/4g, where u is the velocity of projection. If the vertical height is different from u^{2}/4g, then the angle will be different from 45^{∘} and the horizontal distance covered will also be less. Clearly, the span of jump depends upon (i) the initial velocity of the jump, (ii) the angle of projection.**Q.88 A body is thrown horizontally with a velocity υ from a tower of height H. After how much time and at what distance from the base of the tower will the body strike the ground?Answer: **Clearly, Distance from the base of the tower, x=υt=

**Answer:** (i)Yes, because the initial vertical velocity of both the balls is zero, and both will cover the same vertical distance under the same vertical acceleration g. (ii)No, because on striking the ground although their vertical velocities will be same but horizontal velocities will be different. Hence their resultant velocities will be different.**Q.90 A horizontal stream of water leaves an opening in the side of a tank. If the opening is h meters above the ground and the stream hits the ground D meter away, then what is the speed of water as it leaves the tank in terms of g, h and D?**

**Answer: **The steam of water leaving the tank behaves like a horizontal projectile and follows parabolic path. **Q.91 A railway carriage moves over a straight, track with acceleration a. A passenger in the carriage drops a stone. What is the acceleration of the stone w. r. t. the carriage and the earth?**

**Answer:** When the stone is dropped, it falls freely under the acceleration due to gravity g. With respect the earth, the acceleration of the stone is g. Inside the carriage, the stone possesses two accelerations: (i) Horizontal acceleration ′a′ due to the motion of the carriage. (ii) Vertical downward acceleration 'g' due to gravity. The acceleration of the stone w. r. t. the carriage =

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