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Question: The given figure shows a $11.7 \mathrm{ft}$ wide ditch with the approach roads at tan angle of $15^{\circ}$ with the horizontal. With what minimum speed should a motorbike be moving on the roads so that it safely crosses the ditch? Assume that the length of the bike is $5 \mathrm{ft}$ and it leaves the road when the front part runs out of the approach road. Solution: Range to be covered by bike $=11.7+5=16.7 \mathrm{ft}$ $\mathrm{R}=\frac{\mathrm{u}^{2} \sin 2 \theta}{2 \mathrm{~g}}$ $...
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Question: A popular game in Indian villages is goli which is played with small glass balls called golis. The goli of one player is situated at a distance of $2.0 \mathrm{~m}$ from the goli of the second player. The second player has to project his goli by keeping the thumb of the left hand at the place of his goli, holding the goli between his two middle fingers and making the throw. If the projected goli hits the goli of the first player, the second player wins. If the height from which the gol...
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Question: In a soccer practice session the football is kept at the center of the field 40 yards from the $10 \mathrm{ft}$ goalposts. A goal is attempted by kicking the football at a speed of $64 \mathrm{ft} / \mathrm{s}$ at an angle of $45^{\circ}$ to the horizontal. Will the ball reach the goal post? Solution: $y=x \tan \theta-\frac{1}{2} g \frac{x^{2}}{u^{2} \cos ^{2} \theta}$ $=(40 \times 3) \tan 45-\frac{1}{2} \frac{(32)(40 \times 3)^{2}}{(64)^{2}(\cos 45)^{2}}$ $=120-112.4$ $y=7.5 \mathrm{f...
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Question: A ball is thrown at a speed of $40 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ with the horizontal. Find (a) The maximum height reached (c) The range of the ball Take $\mathrm{q}=10 \mathrm{~m} / \mathrm{s}^{2}$ Solution: $\mathrm{u}=40 \mathrm{~m} / \mathrm{s} ; \theta=60^{\circ}$ (a) $\max ^{=} \frac{\mathrm{u}^{2} \sin 2 \theta}{2 \mathrm{~g}}=\frac{40^{2}\left(\sin ^{2} 60\right)}{2 \mathrm{~g}}$ $\mathrm{H}_{\max }=60 \mathrm{~m}$ (b) $\mathrm{R}=\frac{\mathrm{u}^{2} \si...
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Question: A ball is thrown horizontally from a point $100 \mathrm{~m}$ above the ground with a speed of $20 \mathrm{~m} / \mathrm{s}$ Find (a) The time it takes to reach the ground (b) The horizontal distance it travels before reaching the ground (c) The velocity (direction and magnitude) with which it strikes the ground. Solution: $x$-axis $y$-axis $\mathrm{u}_{\mathrm{x}}=20 \mathrm{~m} / \mathrm{s} \mathrm{u}_{\mathrm{y}}=0 \mathrm{~m} / \mathrm{s}$ $a_{x}=0 m / s^{2} a_{y}=g m / s^{2}$ (a) $...
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Question: An elevator is descending with uniform acceleration. To measure the acceleration a person in the elevator drops a coin at the moment the elevator starts. The coin is $6 \mathrm{tt}$ above the floor of the elevator at the time it is dropped, The person observes that the coin strikes the floor in 1 second. Calculate from these data the acceleration of the elevator. Solution: For coin-lift $\mathrm{u}_{\mathrm{re}}=0 \mathrm{~m} / \mathrm{s}$ $\mathrm{t}_{\mathrm{re}}=1 \mathrm{sec}$ $\ma...
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Question: A ball is dropped from a height of $5 \mathrm{~m}$ onto a sandy floor and penetrates the sand up to $10 \mathrm{~cm}$ before coming to rest. Find the retardation of the ball in sand assuming it to be uniform. Solution: For ball in air $\mathrm{u}=0 \mathrm{~m} / \mathrm{s} ; \mathrm{a}=\mathrm{g} ; \mathrm{s}=5 \mathrm{~m}$ $s=u t+\frac{1}{2} a t^{2}$ $5=0+\frac{1}{2}(g) t^{2}$ $t=\sqrt{\frac{10}{g}}$ $\mathrm{v}=\mathrm{u}+\mathrm{at}$ $\mathrm{V}=0+\mathrm{g} \sqrt{\frac{10}{\mathrm{...
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Question: A ball is dropped from a height. If it takes $0.200$ s to cross the last $6.00 \mathrm{~m}$ before hitting the ground, find the height from which it was dropped. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ Solution: For last $6 \mathrm{~m}$, $\mathrm{t}=0.2 \mathrm{sec} ; \mathrm{s}=6 \mathrm{~m} ; \mathrm{a}=\mathrm{g}$ $s=u t+\frac{1}{2} a t^{2}$ $6=\mathrm{u}(0.2)+\frac{\frac{1}{2}}{}(\mathrm{~g})(0.2)^{2}$ $\mathrm{u}=29 \mathrm{~m} / \mathrm{s}$ Before last $6 \mathrm{~m}$, $v^{2}=u^...
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Question: An NCC parade is going at a uniform speed of $6 \mathrm{~km} / \mathrm{h}$ through a place under a berry tree on which a bird is sitting at a height of $12.1 \mathrm{~m}$. A particular instant the bird drops a berry. Which cadet (give the distance from the tree at the instant) will receive the berry on his uniform? Solution: For berry, $\mathrm{u}=0 ; \mathrm{a}=\mathrm{g} ; \mathrm{s}=12.1 \mathrm{~m}$ $\mathrm{s}=u t^{\frac{1}{2}} a t^{2}$ $12.1=0+\frac{1}{2}(g) t^{2}$ $\mathrm{t}=1....
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Question: A healthy young man standing at a distance of $7 \mathrm{~m}$ from a $11: 8 \mathrm{~m}$ high building sees a kid slipping from the top floor. With what speed (assumed uniform) should he run to catch the kid at the arms height $(1.8 \mathrm{~m})$ ? Solution: For kid, $\mathrm{u}=0 ; \mathrm{a}=\mathrm{g} ; \mathrm{s}=11.8-1.8$ $\mathrm{s}=10 \mathrm{~m}$ $s=u t+{ }^{\frac{1}{2}} a t^{2}$ $10=0+\frac{1}{2}(g) t^{2}$ $t=1.42 \mathrm{sec}$ In this time, man has to reach building Speed $=\...
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Question: A person sitting on top of a tall building is dropping balls at regular intervals of one second. Find the positions of the $3^{\text {rd }}, 4^{\text {th }}$ and $5^{\text {th }}$ ball when the $6^{\text {th }}$ ball is being dropped. Solution: For every ball; $u=0$ and $a=g$ When $6^{\text {th }}$ ball is dropped, $5^{\text {th }}$ ball moves for 1 second, $4^{\text {th }}$ ball moves for 2 seconds, $3^{\text {rd }}$ ball moves for 3 seconds Position $\mathrm{S}=u t^{\frac{1}{2}} a t^...
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Question: A stone is thrown vertically upward with a speed of $28 \mathrm{~m} / \mathrm{s}$. (a) Find the maximum height reached by the stone. (b) Find its velocity one second before it reaches the maximum height. (c) Does the answer pf part (b) change if the initial speed is more than $28 \mathrm{~m} / \mathrm{s}$ such as $40 \mathrm{~m} / \mathrm{s}$ or 80 $\mathrm{m} / \mathrm{s}$ ? Solution: (a) $u=28 \mathrm{~m} / \mathrm{s} ; \mathrm{v}=0 \mathrm{~m} / \mathrm{s} ; \mathrm{a}=-\mathrm{g}$ ...
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Question: A ball is dropped from a balloon going up at a speed of $7 \mathrm{~m} / \mathrm{s}$. if the balloon was at a height $60 \mathrm{~m}$ at the time of dropping the ball, how long will the ball take in reaching the ground? Solution: $\mathrm{u}=7 \mathrm{~m} / \mathrm{s} ; \mathrm{a}=-\mathrm{g} ; \mathrm{s}=-60$ $s=u t+\frac{1}{2} a t^{2}$ $-60=7 t^{\frac{1}{2}}(g) t^{2}$ $\mathrm{t}=4.28 \mathrm{sec}$...
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Question: A ball is projected vertically upward with a speed of $50 \mathrm{~m} / \mathrm{s}$ Find (a) The maximum height (b) The time to reach the maximum height (c) The speed at half the maximum height. Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ Solution: $\mathrm{u}=50 \mathrm{~m} / \mathrm{s} ; \mathrm{v}=0 \mathrm{~m} / \mathrm{s} ; \mathrm{a}=-\mathrm{g}$ (a) $\mathrm{v}^{2}=\mathrm{u}^{2}+2 \mathrm{as}$ $0^{2}=(50)^{2}+2(\mathrm{~g}) \mathrm{s}$ $\mathrm{s}=125 \mathrm{~m}$ (b) $v=...
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Question: A car travelling at $60 \mathrm{~km} / \mathrm{h}$ overtakes another car travelling at $42 \mathrm{~km} / \mathrm{h}$. Assuming each car to be $5.0 \mathrm{~m}$ long, find the time taken during the overtake and the total road distance used for the overtake. Solution: $\vec{V}_{1}=60^{\times \frac{5}{18}}=16.6 \mathrm{~m} / \mathrm{s}$ $\vec{V}_{2}=42^{\times \frac{2}{18}}=11.6 \mathrm{~m} / \mathrm{s}$ Relative velocity $=16.6-11.6$ $V_{\text {rel }}=5 \mathrm{~m} / \mathrm{s}$ $\mathr...
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Question: A police jeep is chasing a culprit going on a motorbike. The motorbike crosses a turning at a speed of $72 \mathrm{~km} / \mathrm{h}$. The jeep follows it at a speed of $90 \mathrm{~km} / \mathrm{h}$, crossing the turning ten seconds later than the bike. Assuming that they travel at constant speeds, how far from the turning will the jeep catch up with the bike? Solution: $V_{\text {bike }}=72^{\times \frac{5}{18}}=20 \mathrm{~m} / \mathrm{s}$ $\mathrm{V}_{\text {police }}=90^{\times \f...
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Question: A driver takes $0.20$ s to apply the brakes after he sees a need for it, This is called the reaction time of the driver. If he is driving a car at a speed of $54 \mathrm{~km} / \mathrm{h}$ and the brakes cause a deceleration of $6.0 \mathrm{~m} / \mathrm{s}^{2}$, find the distance travelled by the car after he sees the need to put the brakes on. Solution: Speed of car $=54 \times 18=15 \mathrm{~m} / \mathrm{s}$ Distance travelled during reaction time $\mathrm{S}_{1}=\mathrm{v}^{\mathrm...
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Question: A particle starting from rest moves with constant acceleration. If it takes $5.0$ s to reach the speed $18.0$ $\mathrm{km} / \mathrm{h}$ Find (a) The average velocity during this period (b) The distance travelled by the particle during this period. Solution: $u=0 ; t=5 s e c ; v=18 \times 5 / 18=5 \mathrm{~m} / \mathrm{s}$ $\mathrm{v}=\mathrm{u}+\mathrm{at}$ $5=0+a(5)$ $a=1 \mathrm{~m} / \mathrm{s}^{2}$ $S=u t+\frac{1}{2} a t^{2}$ $\mathrm{S}=0+\frac{1}{2}(\mathrm{a})(5)^{2}$ $\mathrm{...
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Question: A bullet going with speed $350 \mathrm{~m} / \mathrm{s}$ enters a concrete wall and penetrates a distance $5.0 \mathrm{~cm}$ before coming to rest. Find the deceleration. Solution: $\mathrm{u}=350 \mathrm{~m} / \mathrm{s} ; \mathrm{v}=0 ; \mathrm{s}=5 \times 10^{-2} \mathrm{~m}$ $v^{2}=2^{2}+2 a s$ $0^{2}=(350)^{2}+2(a)(0.05)$ $\mathrm{a}=-12.25 \times 10^{5} \mathrm{~m} / \mathrm{s}^{2}$...
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Question: A bullet travelling with a velocity of $16 \mathrm{~m} / \mathrm{s}$ penetrates a tree trunk and comes to rest in $0.4 \mathrm{~m}$. Find the time taken during the retardation. Solution: $\mathrm{u}=16 \mathrm{~m} / \mathrm{s} ; \mathrm{v}=0 \mathrm{~m} / 2 ; \mathrm{s}=0.4 \mathrm{~m}$ $v^{2}=u^{2}+2 a s$ $0^{2}=(1)^{2}+2(a)(0.4)$ $a=-320 m / s^{2}$ $\mathrm{v}=\mathrm{u}+$ at $0=16+(-320) t$ $t=0.05 \mathrm{sec}$...
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Question: A train starts from rest and moved with a constant acceleration of $2.0 \mathrm{~m} / \mathrm{s}^{2}$ for half a minute. The brakes are then applied and the train comes to rest in one minute. (a) Find the total distance moved by the train (b) The maximum speed attained by the train (c) The position(s) of the train at half the maximum speed. Solution: A $=$ slope $=\frac{v}{t}$ $2=\frac{v}{30}$ $\mathrm{v}=60 \mathrm{~m} / \mathrm{s}$ (a) Distance=Area of v-t graph $=\frac{1^{\prime}}{2...
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Question: A person travelling at $43.2 \mathrm{~km} / \mathrm{h}$ applies the brake giving a deceleration of $6.0 \mathrm{~m} / \mathrm{s}^{2}$ to his scooter. How far will it travel before stopping? Solution: $\mathrm{u}=43.2^{\times \frac{5}{18}}=12 \mathrm{~m} / \mathrm{s} ; \mathrm{v}=0 ; \mathrm{a}=-6 \mathrm{~m} / \mathrm{s}^{2}$ Using, $\mathrm{v}^{2}=\mathrm{u}^{2}+2 \mathrm{as}$ $\mathrm{O}^{2}=(12)^{2}-2^{X^{X_{s}}} 6^{\mathrm{s}}$ $\mathrm{s}=12 \mathrm{~m}$...
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Question: An object having a velocity $4.0 \mathrm{~m} / \mathrm{s}$ is accelerated at the rate of $1.2 \mathrm{~m} / \mathrm{s}^{2}$ for $5.0 \mathrm{~s}$. Find the distance travelled during the period of acceleration. Solution: $\mathrm{u}=4 \mathrm{~m} / \mathrm{s} ; \mathrm{a}=1.2 \mathrm{~m} / \mathrm{s}^{2} ; \mathrm{t}=5 \mathrm{sec}$. Distance travelled $s=u t+\frac{1}{2} a t^{2}$ $s=(4)(5)+\frac{\frac{1}{2}}{2}(1.2)(5)^{2}$ $=35 \mathrm{~m}$...
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Question: A particle starts from a point $A$ and travels along the solid curve shown in the given figure. Find approximately the position B of the particle such that the average velocity between the positions $A$ and $B$ has the same direction as the instantaneous velocity at $B$. Solution: Direction of instantaneous velocity of point $B$ must be same as direction of average velocity $\overrightarrow{A B}$. So, point is approximately $(5,3)$...
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Question: The given figure shows $x-t$ graph of a particle. Find the time $t$ such that the average velocity of the particle during the period 0 to $t$ is zero. Solution: Average velocity is zero when displacement is zero At $t=0 ; x=20$ and again at $t=12 ; x=20$...
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