A monkey climbs up a slippery pole for 3 seconds
Question: A monkey climbs up a slippery pole for 3 seconds and subsequently slips for 3 seconds. Its velocity at time t is given by v(t) = 2t (3 t); 0t3 and v(t) = -(t 3) ( 6 t) for 3 t 6s on m/s. It repeats this cycle till it reaches the height of 20 m. (a) At what time is its velocity maximum? (b) At what time is its average velocity maximum? (c) At what times is its acceleration maximum in magnitude? (d) How many cycles are required to reach the top? Solution: (a) For maximum velocity v(t) dv...
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Question: Prove that $\tan \left(\frac{\pi}{4}+x\right)=\frac{1+\tan x}{1-\tan x}$ Solution: In this question the following formulas will be used: $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$ $\tan \left(\frac{\pi}{4}+x\right)=\frac{\tan \frac{\pi}{4}+\tan x}{1-\tan \frac{\pi}{4} \tan x}$ $=\frac{1+\tan x}{1-\tan x} \because \tan \frac{\pi}{4}=1$...
Read More →A motor car moving at a speed of 72 km/h
Question: A motor car moving at a speed of 72 km/h cannot come to a stop in less than 3 s while for a truck this time interval is 5 s. On a highway the car is behind the truck both moving at 72 km/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it does not bump onto the truck. Human response time is 0.5 s. Solution: For truck, u = 20 m/s v = 0 a = ? t = 5s v = u + at a = 4 m/s2 For car, t = 3 s u = 20 m/s v = 0 a = ac v...
Read More →It is a common observation that rain clouds
Question: It is a common observation that rain cloudscan be at about a kilometre altitude above the ground. (a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also, calculate in km/h (b) A typical rain drop is about 4 mm diameter. Momentum is mass x speed in magnitude. Estimate its momentum when it hits ground. (c) Estimate the time required to flatten the drop. (d) Rate of change of momentum is force. Estimate how much force such a drop would exert on you....
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Question: Prove that $\sin \left(x-\frac{\pi}{6}\right)+\cos \left(x-\frac{\pi}{3}\right)=\sqrt{3} \sin x$ Solution: In this question the following formulas will be used: $\sin (A-B)=\sin A \cos B-\cos A \sin B$ $\cos (A-B)=\cos A \cos B+\sin A \sin B$ $=\sin x \cos \frac{\pi}{6}-\cos x \sin \frac{\pi}{6}+\cos x \cos \frac{\pi}{3}+\sin x \sin \frac{\pi}{3}$ $=\sin x \times \frac{\sqrt{3}}{2}-\cos x \times \frac{1}{2}+\cos x \times \frac{1}{2}+\sin x \times \frac{\sqrt{3}}{2}$ $=\sin x \times \fr...
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Question: Prove that $\cos x+\cos \left(120^{\circ}-x\right)+\cos \left(120^{\circ}+x\right)=0$ Solution: In this question the following formulas will be used: $\cos (A+B)=\cos A \cos B-\sin A \sin B$ $\cos (A-B)=\cos A \cos B+\sin A \sin B$ $=\cos x+\cos 120^{\circ} \cos x-\sin 120 \sin x+\cos 120^{\circ} \cos x+\sin 120 \sin x$ $=\cos x+2 \cos 120 \cos x$ $=\cos x+2 \cos (90+30) \cos x$ $=\cos x+2(-\sin 30) \cos x$ $=\cos x-2 \times \frac{1}{2} \cos x$ $=\cos x-\cos x$ $=0$....
Read More →A ball is dropped from a building of height 45 m.
Question: A ball is dropped from a building of height 45 m. Simultaneously another ball is thrown up with a speed 40 m/s. Calculate the relative speed of the balls as a function of time. Solution: V = v1= ? U = 0 h = 45 m a = g t = t V = u + at v1= 0 + gt v1= gt Therefore, when the ball is thrown upward, v1= -gt V = v2 u = 40 m/s a = g t = t V = u + at v2= 40 gt The relative velocity of the ball in the downward direction is 40 m/s But when the speed increases due to acceleration, the relative sp...
Read More →It is given that for the function
Question: It is given that for the function $f(x)=x^{3}-6 x^{2}+a x+b$ on $[1,3]$, Rolle's theorem holds with $c=2+\frac{1}{\sqrt{3}} .$ If $f(1)=f(3)=0$, thena=_______,b =________. Solution: The given function is $f(x)=x^{3}-6 x^{2}+a x+b$. It is given that Rolle's theorem holds for $f(x)$ defined on $[1,3]$ with $c=2+\frac{1}{\sqrt{3}}$. $f(1)=f(3)=0 \quad$ (Given) $\therefore f(1)=0$ $\Rightarrow 1-6+a+b=0$ $\Rightarrow a+b=5$ ......(1) Also, $\Rightarrow 27-54+3 a+b=0$ $\Rightarrow 3 a+b=27$...
Read More →It is given that for the function
Question: It is given that for the function $f(x)=x^{3}-6 x^{2}+a x+b$ on $[1,3]$, Rolle's theorem holds with $c=2+\frac{1}{\sqrt{3}} .$ If $f(1)=f(3)=0$, thena=_______,b =________. Solution: The given function is $f(x)=x^{3}-6 x^{2}+a x+b$. It is given that Rolle's theorem holds for $f(x)$ defined on $[1,3]$ with $c=2+\frac{1}{\sqrt{3}}$. $f(1)=f(3)=0 \quad$ (Given) $\therefore f(1)=0$ $\Rightarrow 1-6+a+b=0$ $\Rightarrow a+b=5$ ......(1) Also, $\Rightarrow 27-54+3 a+b=0$ $\Rightarrow 3 a+b=27$...
Read More →A bird is tossing between two cars moving towards
Question: A bird is tossing between two cars moving towards each other on a straight road. One car has a speed of 18 m/h while the other has the speed of 27 km/h. The bird starts moving from first car towards the other and is moving with the speed of 36 km/h and when the two cars were separated by 36 km. What is the total distance covered by the bird? What is the total displacement of the bird? Solution: The relative speed of the cars = 27 + 18 = 45 km/h When the two cars meet together, time t i...
Read More →Prove that
Question: Prove that $\sin \left(150^{\circ}+x\right)+\sin \left(150^{\circ}-x\right)=\cos x$ Solution: In this question the following formula will be used: $\sin (A+B)=\sin A \cos B+\cos A \sin B$ $\sin (A-B)=\sin A \cos B-\cos A \sin B$ $=\sin 150^{\circ} \cos x+\cos 150^{\circ} \sin x+\sin 150^{\circ} \cos x-\cos 150^{\circ} \sin x$ $=2 \sin 150^{\circ} \cos x$ $=2 \sin \left(90^{\circ}+60^{\circ}\right) \cos x$ $=2 \cos 60^{\circ} \cos x$ $=2 \times \frac{1}{2} \cos x$ $=\cos x$...
Read More →A particle executes the motion described by
Question: A particle executes the motion described by x(t) = x0(1 e-t) where t 0, x0 0 (a) Where does the particles start and with what velocity? (b) Find maximum and minimum values of x(t), v(t), a(t). Show that x(t) and a(t) increase with time and v(t) decreases with time. Solution: (a) x(t) = x0(1 e-t) v(t) = dx(t)/dt =+x0 e-t a(t) = dv/dt = x02e-t v(0) = x0 (b) x(t) is minimum at t = 0 since t = 0 and [x(t)]min = 0 x(t) is maximum at t = since t = and [x(t)]max = e-t = v(t) is maximum at t =...
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Question: Prove that (i) $2 \sin \frac{5 \pi}{12} \sin \frac{\pi}{12}=\frac{1}{2}$ (ii) $2 \cos \frac{5 \pi}{12} \cos \frac{\pi}{12}=\frac{1}{2}$ (iii) $2 \sin \frac{5 \pi}{12} \cos \frac{\pi}{12}=\frac{(2+\sqrt{3)}}{2}$ Solution: (i) $2 \sin \frac{5 \pi}{12} \cdot \sin \frac{\pi}{12}=-\left(\cos \left(\frac{5 \pi}{12}+\frac{\pi}{12}\right)-\cos \left(\frac{5 \pi}{12}-\frac{\pi}{12}\right)\right)$ [Using 2sinx.siny = cos(x + y)cos (xy)] $=-\left(\cos \frac{6 \pi}{12}-\cos \frac{4 \pi}{12}\right)...
Read More →An object falling through a fluid is observed
Question: An object falling through a fluid is observed to have acceleration given by a = g bv where g = gravitational acceleration and b is constant. After a long time of release, it is observed to fall with constant speed. What must be the value of constant speed? Solution: The concept used in this question will be based on the behaviour of a spherical object when it is dropped through a viscous fluid. When a spherical body of radius r is dropped, it is first accelerated and gradually the acce...
Read More →Give example of a motion where
Question: Give example of a motion where x0, v0, a0 at a particular instant. Solution: Let the motion be represented as: x(t) = A + Bet Let AB and0 Velocity is x(t) = dx/dt = -Bet Acceleration is a(t) = dx/dt = B2et Therefore, it can be said that x(t) 0, v(t) 0, and a0...
Read More →If the function
Question: If the function $f(x)=x^{3}-6 x^{2}+a x+b$ defined on $[1,3]$ satisfies Roll's theorem for $c=2+\frac{1}{\sqrt{3}}$, then $a=$___________,b = __________. Solution: The given function is $f(x)=x^{3}-6 x^{2}+a x+b$. It is given that $f(x)$ defined on $[1,3]$ satisfies Rolle's theorem for $c=2+\frac{1}{\sqrt{3}}$. $\therefore f(1)=f(3)$ and $f^{\prime}(c)=0$ Now, $f(1)=f(3)$ $\Rightarrow 1-6+a+b=27-54+3 a+b$ $\Rightarrow-5+a=-27+3 a$ $\Rightarrow 2 a=22$ $\Rightarrow a=11$ Also, $f(x)=x^{...
Read More →Give examples of a one-dimensional motion where
Question: Give examples of a one-dimensional motion where (a) the particle moving along positive x-direction comes to rest periodically and moves forward (b) the particle moving along positive x-direction comes to rest periodically and moves backward Solution: When an equation has sine and cosine functions, the nature is periodic. (a) When the particle is moving in positive x-direction, it is given as t sin t When the displacement is as a function of time, it is given as x(t) = t sin t When the ...
Read More →A ball is bouncing elastically with a speed 1 m/s
Question: A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground, (a) the direction of motion of the ball changes every 10 seconds (b) speed of ball changes every 10 seconds (c) average speed of ball over any 20 seconds intervals is fixed (d) the acceleration of ball is the same as from ...
Read More →Prove that
Question: Prove that (i) $\cos \left(\frac{\pi}{3}+x\right)=\frac{1}{2}(\cos x-\sqrt{3} \sin x)$ (ii) $\sin \left(\frac{\pi}{4}+x\right)+\sin \left(\frac{\pi}{4}-x\right)=\sqrt{2} \cos x$ (iii) $\frac{1}{\sqrt{2}} \cos \left(\frac{\pi}{4}+x\right)=\frac{1}{2}(\cos x-\sin x)$ (iv) $\cos \mathrm{x}+\cos \left(\frac{2 \pi}{3}+\mathrm{x}\right)+\cos \left(\frac{2 \pi}{3}-\mathrm{x}\right)=0$ Solution: (i) $\cos \left(\left(\frac{\pi}{3}+x\right)=\cos \frac{\pi}{3} \cdot \cos x-\sin \frac{\pi}{3} \cd...
Read More →A spring with one end attached to a mass
Question: A spring with one end attached to a mass and the other to a rigid support is stretched and released. (a) magnitude of acceleration, when just released is maximum (b) magnitude of acceleration, when at equilibrium position is maximum (c) speed is maximum when mass is at equilibrium position (d) magnitude of displacement is always maximum whenever speed is minimum Solution: The correct answer is (a) magnitude of acceleration, when just released is maximum and (c) speed is maximum when ma...
Read More →For the function
Question: For the function $f(x)=8 x^{2}-7 x+5, x \in[-6,6]$, the value of $c$ for the lagrange's mean value theorem is ______________ Solution: The given function is $f(x)=8 x^{2}-7 x+5$. f(x) is a polynomial function. We know that a polynomial function is everywhere continuous and differentiable. So, $f(x)$ is continuous on $[-6,6]$ and differentiable on $(-6,6)$. Thus, both the conditions of Lagrange's mean value theorem are satisfied. So, there must exist at least one real number $c \in(-6,6...
Read More →For the function
Question: For the function $f(x)=8 x^{2}-7 x+5, x \in[-6,6]$, the value of $c$ for the lagrange's mean value theorem is ______________ Solution: The given function is $f(x)=8 x^{2}-7 x+5$. f(x) is a polynomial function. We know that a polynomial function is everywhere continuous and differentiable. So, $f(x)$ is continuous on $[-6,6]$ and differentiable on $(-6,6)$. Thus, both the conditions of Lagrange's mean value theorem are satisfied. So, there must exist at least one real number $c \in(-6,6...
Read More →For the one-dimensional motion,
Question: For the one-dimensional motion, describe by x = t sint (a) x(t)0 for all t0 (b) v(t)0 for all t0 (c) a(t)0 for all t0 (d) v(t) lies between 0 and 2 Solution: The correct answer is (a) x(t)0 for all t0 and d) v(t) lies between 0 and 2...
Read More →Solve this
Question: If $\cos \mathrm{x}=\frac{3}{5}$ and $\cos \mathrm{y}=\frac{-24}{25}$, where $\frac{3 \pi}{2}\mathrm{x}2 \pi$ and $\pi\mathrm{y}\frac{3 \pi}{2}$, find the values of (i) $\sin (x+y)$ (ii) $\cos (x-y)$ (iii) $\tan (x+y)$ Solution: Given $\cos x=\frac{3}{5}$ and $\cos y=\frac{-24}{25}$ We will first find out value of sinx and siny $\sin x=\sqrt{\left(1-\cos ^{2} x\right)} \Rightarrow \sqrt{\left(1-\left(\frac{3}{5}\right)^{2}\right)}=\sqrt{\left(\frac{25-9}{25}\right)} \Rightarrow \sqrt{\...
Read More →At a metro station,
Question: At a metro station, a girl walks up a stationary escalator in time t1. If she remains stationary on the escalator, then the escalator take her up in time t2. The time taken by her to walk up on the moving escalator will be (a) (t1+ t2)/2 (b) t1t2/(t2 t1) (c) t1t2/(t2+ t1) (d) t1 t2 Solution: The correct answer is (c) t1t2/(t2+ t1)...
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