Motion in A Straight Line Test

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Motion in A Straight Line Test - 71

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Question 1
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A car starts from rest and moves in a straight line with a constant acceleration $$\alpha$$. After time $$t_0$$, brakes are applied which causes retardation of magnitude $$\beta$$ and car finally stops. The distance travelled by the car is

A
$$\alpha t_0^2 \cfrac{(\alpha + \beta)}{4\beta}$$

B
$$\alpha t_0^2 \cfrac{(\alpha + \beta)}{2\beta}$$

C
$$\beta t_0^2 \cfrac{(\alpha + \beta)}{2\alpha}$$

D
$$\beta t_0^2 \cfrac{(\alpha + \beta)}{4\alpha}$$

Solution

Distance traveled in $$t_0$$ seconds
$$S_1=0\times t_0+\dfrac{1}{2}\alpha t^2$$(u=0 which is at rest)
Final velocity after $$t_0$$ seconds,
$$V_s=0+\alpha t_0$$
Final velocity after applying retardation $$\beta=0$$
Initial velocity =final velocity after $$t_0$$ seconds
$$0=(V_s)^2-2\beta(s_2)$$ ($$s_2$$ is the distance traveled after $$\beta$$ retardation)
$$s_2=\dfrac{(V_s)^2}{2\beta}$$
$$s_2=\dfrac{\alpha^2t_0^2}{2\beta}$$
$$s_1+s_2=\dfrac{1}{2}\alpha t_0^2(\alpha+\dfrac{\alpha}{\beta})$$
$$=\dfrac{\alpha t_0^2(\alpha+\beta)}{2 \beta}$$
Question 2
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A horse is running with constant acceleration $$\dfrac {g}{\sqrt {3}}$$. A small ball is projected by a horce rider at an angle $$\theta$$ with horizontal with respect to house.The value of $$\theta$$ such that the ball again caught by the boy

A
$$36^{o}$$

B
$$45^{o}$$

C
$$60^{o}$$

D
$$90^{o}$$

Solution
Question 3
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A lift whose cage is $$3\ m$$ high is moving up with an acceleration of $$2\ m/s^{2}$$. A piece of stone is dropped from the top of the cage of the lift when its velocity is $$8\ m/s$$. If $$g = 10\ m/s^{2}$$, then the stone will reach the floor of the lift after.

A
$$0.7\ s$$

B
$$0.5\ s$$

C
$$0.4\ s$$

D
$$0.3\ s$$

Solution

The time taken to reach the stone on the floor is given as,

$$s = ut + \frac{1}{2}\left( {a + g} \right){t^2}$$

$$3 = 8 \times t + \frac{1}{2}\left( {2 + 10} \right){t^2}$$

$$3 = 8 \times t + 6{t^2}$$

$$t = 0.3052\;\sec $$
Question 4
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A wooden plank of mass 20kg is resting on a smooth horizontal floor. A man of mass 60kg starts moving from one end of the plank to the other end. The length of the plank is 10m. Find the displacement of the plank over the floor when the man reaches the other end of the plank.

A
5m

B
6.5m

C
2.5m

D
7.5m

Solution
Question 5
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A trolley was moving horizontally on a smooth ground with velocity $$v$$ with respect to the earth. Suddenly a man starts running from rear end of the trolley with a velocity $$\dfrac {3v}{2}$$ with respect to the trolley in opposite direction. If the length of the trolley is $$L$$, find the displacement of the man with respect to earth when he reaches the starting point on the trolley. [Mass if the trolley is equal to the mass of the man].

A
$$\dfrac {L}{3}$$

B
$$\dfrac {2L}{3}$$

C
$$\dfrac {4L}{3}$$

D
$$zero$$

Solution
Question 6
1 / -0

water drop are falling from calling at regular interval of time in such away that sixth drop if going to be deleched when $$1^{st}$$ drop it just hitting the ground. If time interval between two consecitive drope is $$0.5$$ second. Then the height of celling it.

A
$$31.25m$$

B
$$45m$$

C
$$62.5m$$

D
$$20m$$

Solution
Question 7
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A force of $$50$$ $$N$$ is required to push a car on a level road with constant speed of $$10$$ $${m/s}$$. The mass of the car is $$500$$ $$kg$$. What force should be applied to make the car accelerate at $$1$$ $${m/s^2}$$?

A
$$550$$ $$N$$

B
$$450$$ $$N$$

C
$$500$$ $$N$$

D
$$2500$$ $$N$$

Solution

Since $$50\;{\rm{N}}$$ force is required for the car to move with constant velocity.

Additional force applied to move a car of $$500\;{\rm{kg}}$$ with acceleration of $$1\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}$$is given by

$$F = ma$$

$$F = 500 \times 1$$

$$F = 500\;{\rm{N}}$$

Thus total force applied

$${F_T} = 500 + 50$$

$$F = 550\;{\rm{N}}$$
Question 8
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Determine in time in which the smaller block reaches other end of bigger block in the figure.

A
$$4\ s$$

B
$$8$$

C
$$2.19\ s$$

D
$$2.13\ s$$

Solution

Both block will not move together because friction force between block is not sufficient to accelerate 8Kg block with same acceleration

Drawing $$FBD$$ of $$2kg$$ block
$$f_r=$$ frictional force
$$f_r=\mu Mg$$
for vertical balance, $$M=20\ N$$

$$fr=0.3\times 20=6\ N$$

Now let the block moves with acceleration $$a_{1}$$
$$\Rightarrow f-fr=ma_{1}$$
$$10-6=2\times a_{1}$$
$$\Rightarrow a_{1}=2\ m/\sec^{2}$$

Drawing $$FBD$$ of block of $$8\ kg$$ mass
$$\Rightarrow$$ Let $$a_{2}$$ be its acceleration

$$a_2=\dfrac{f_r}{m_2}=\dfrac{.3\times 20}{8}$$

$$a_{2}=\dfrac{6}{8}=\dfrac{3}{4}\ m/\sec^{2}$$

Relative acceleration of both the blocks
$$a=a_{1}-a_{2}=2-\dfrac{3}{4}=\dfrac{5}{4}$$

distance to be covered $$=3\ m$$ initial velocity $$=0$$
$$S=ut+\dfrac{1}{2}at^{2}$$

$$=3=0+\dfrac{1}{2}\times \dfrac{5}{4}\times t^{2}$$

$$\Rightarrow t=2.19\ \sec$$

Option $$C$$
Question 9
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A body moving with uniform acceleration along a straight line covers $$21\ m$$ in the fifth seconds of its motion and $$41\ m$$ in the tenth seconds of its motion. What is its initial velocity?

A
$$2\ m/s$$

B
$$3\ m/s$$

C
$$4\ m/s$$

D
$$5\ m/s$$

Solution
Question 10
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A car is travelling on a straight road. The maximum velocity the car can attain is $$24\ ms^{-1}$$. The maximum acceleration and deceleration it can attain are $$1\ ms^{-2}$$ and $$4\ ms^{-2}$$ respectively. The shortest time the car takes to start from rest and come to rest in a distance of $$200$$ meter is

A
$$22.4\ second$$

B
$$30.0\ second$$

C
$$11.2\ second$$

D
$$5.6\ second$$

Solution