Motion in A Straight Line Test

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Motion in A Straight Line Test - 67

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Question 1
1 / -0

On a horizontal flat ground, a person is standing at a point $$A$$. At this point, he installs a $$5 m$$ long pole vertically. Now, he moves $$5 m$$ towards east and then $$2 m$$ towards pole to the top of the second pole. Find the displacement and magnitude of the displacement of the bird.

A
$$5\hat{i}+2\hat{j}-2\hat{k}$$ ; $$\sqrt{33}m$$

B
$$6\hat{i}+2\hat{j}-2\hat{k}$$ ; $$\sqrt{23}m$$

C
$$5\hat{i}+3\hat{j}-2\hat{k}$$ ; $$\sqrt{33}m$$

D
$$5\hat{i}+2\hat{j}-2\hat{k}$$ ; $$\sqrt{23}m$$

Solution
Question 2
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Directions For Questions

An elevator without a ceiling is ascending up with an acceleration up with an acceleration of $$5\ ms^{-2}$$. A boy on the elevator shoots a ball in vertically upward direct from a height $$2\ m$$ above the floor of elevator. At this instant, the elevator is moving up with a velocity of $$10\ ms^{-1}$$ and floor, of the elevator is at a height of $$50\ m$$ from the ground. The initial speed of the ball is $$15\ ms^{-1}$$ w.r.t. the elevator. Consider the duration for which the ball strikes the floor of the elevator in answering following question:

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The distance travelled by the ball during its flight is:

A
$$32.64\ m$$

B
$$31.86\ m$$

C
$$52\ m$$

D
$$30.56\ m$$

Solution

Let initial point of floor of elevator be the reference point.
We have, $$x={ x }_{ 0 }+ut+\dfrac { 1 }{ 2 } { at }^{ 2 }$$
$$\therefore$$ $${ x }_{ ball }=2+25t-\dfrac { 1 }{ 2 } \times 10\times { t }^{ 2 }\quad \rightarrow \left( 1 \right) $$
$${ x }_{ elevator }=0+10t+5{ t }^{ 2 }\quad \rightarrow \left( 2 \right) $$
The ball will hit the floor when
$${ x }_{ ball }={ x }_{ elevator }$$
On solving $$(1)$$ & $$(2)$$, $$t=2.13s$$
$$\therefore$$ Distance traveled by the ball during height is,
$$S=ut+\dfrac { 1 }{ 2 } { at }^{ 2 }$$
$$=25\times 2.13-\dfrac { 1 }{ 2 } \times 10\times { 2.13 }^{ 2 }$$
$$=30.56m$$
Question 3
1 / -0

A particle of $$m = 5 kg$$ is momentarily at rest at $$x = 0$$ at $$t = 0$$. It is acted upon by two forces$$\vec{F}_{1}$$ and $$\vec{F}_{2}$$. $$\vec{F}_{1} = 70 \hat{j} N$$. The direction and magnitude of $$\vec{F}_{2}$$ are unknown. The particle experiences a constant acceleration, a, in the direction as shown in figure. Neglect gravity.
a. Find the missing force $$\vec{F}_{2}$$.
b. What is the velocity vector of the particle at $$t=10s$$?
c. What third force, $$\vec{F}_{3}$$, is required to make the acceleration of the particle zero? Either give magnitude and direction of $$\vec{F}_{3}$$ or its components.

A
a. $$30\sqrt{2}$$
b. $$60\hat{i} + 80\hat{j}$$
c. $$30\hat{i} - 40\hat{j}$$

B
a. $$30\sqrt{2}$$
b. $$60\hat{i} + 90\hat{j}$$
c. $$30\hat{i} - 40\hat{j}$$

C
a. $$30\sqrt{2}$$
b. $$60\hat{i} + 80\hat{j}$$
c. $$30\hat{i} - 50\hat{j}$$

D
a. $$30\sqrt{2}$$
b. $$60\hat{i} + 80\hat{j}$$
c. $$40\hat{i} - 40\hat{j}$$

Solution
Question 4
1 / -0

A ship is sailing due north at a speed of $$1.25$$ m s$$^{-1}$$. The current is taking it towards east at the rate of $$2$$ m s$$^{-1}$$ and a sailor is climbing a vertical pole in the ship at the rate of $$0.25$$ m s$$^{-1}$$. Find the magnitude of the velocity of the sailor with respect to ground.

A
$$=\dfrac{3}{2}\sqrt{\dfrac{5}{2}}ms^{-1}$$

B
$$=\dfrac{3}{22}\sqrt{\dfrac{5}{2}}ms^{-1}$$

C
$$=\dfrac{23}{2}\sqrt{\dfrac{5}{2}}ms^{-1}$$

D
$$=\dfrac{13}{2}\sqrt{\dfrac{5}{2}}ms^{-1}$$

Solution
Question 5
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$$B_1$$, $$B_2$$, and $$B_3$$ are three balloons ascending with velocities $$v, 2v$$, and $$3v$$, respectively. If a bomb is dropped from each when they are at the same height, then

A
Bomb from $$B_1$$ reaches the ground first

B
Bomb from $$B_2$$ reaches the ground first

C
Bomb from $$B_3$$ reaches the ground first

D
They reach the ground simultaneously

Solution
Question 6
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A flat car of mass $$M$$ is at rest on a frictionless floor with a child of mass $$m$$ standing at its edge. If the child jumps off from the car toward right with an initial velocity $$u$$, with respect to the car, velocity of the car after its jump.

A
$$\nu = \dfrac{mu}{2m + M}$$

B
$$\nu = \dfrac{mu}{m + M}$$

C
$$\nu = \dfrac{2mu}{3m + M}$$

D
$$\nu = \dfrac{mu}{m + 4M}$$

Solution
Question 7
1 / -0

A particle moves along a straight line such that its displacement S varies with time t as S = a + bt + g$$t^2$$
Column I
Column II
i Acceleration at t = 2 s a $$\beta + 5 \gamma$$
ii Average velocity during third second b $$2\gamma$$
iii Velocity at t = 1 s c $$\alpha$$
iv Initial displacement d $$\beta = 2 \gamma$$

A
(i) - b, (ii) - a, (iii) - d, (iv) - d.

B
(i) - b, (ii) - a, (iii) - d, (iv) - c.

C
(i) - a, (ii) - b, (iii) - d, (iv) - c.

D
(i) - b, (ii) - c, (iii) - d, (iv) - a.

Solution
Question 8
1 / -0

In a car race, car A takes 4 s less than car B at the finish and passes the finishing point with a velocity v more than the car B. Assuming that the cars start from rest and travel with constant accelerations $$a_1 = 4 ms^{-2}$$and $$a_2 = 1 ms^{-2}$$ respectively, find the velocity of v in m/s.

A
9

B
8

C
10

D
11

Solution
Question 9
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A cat, on seeing a rat at a distance d = 5 m, starts with velocity u = 5 m/s and moves with the acceleration $$\alpha = 2.5 ms^{-2}$$ in order to catch it, while the rate with the acceleration $$\beta$$ starts from rest. For what value of $$\beta$$ will the cat overtake the rat? (in $$ms^{-2}$$)

A
5

B
6

C
7

D
8

Solution
Question 10
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A ball is thrown downwards with a speed of 20 m/s from the top of a building 150 m high and simultaneously another ball is thrown vertically upwards with a speed of 30 m/s from the foot of the building. Find the time after which both the balls will meet. (g = 10 $$ms^{-2}$$)

A
33

B
3

C
4

D
5

Solution

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	Column I		Column II
i	Acceleration at t = 2 s	a	$$\beta + 5 \gamma$$
ii	Average velocity during third second	b	$$2\gamma$$
iii	Velocity at t = 1 s	c	$$\alpha$$
iv	Initial displacement	d	$$\beta = 2 \gamma$$

Motion in A Straight Line Test - 67

Result

Motion in A Straight Line Test - 67

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Question 1

$$5\hat{i}+2\hat{j}-2\hat{k}$$ ; $$\sqrt{33}m$$

$$6\hat{i}+2\hat{j}-2\hat{k}$$ ; $$\sqrt{23}m$$

$$5\hat{i}+3\hat{j}-2\hat{k}$$ ; $$\sqrt{33}m$$

$$5\hat{i}+2\hat{j}-2\hat{k}$$ ; $$\sqrt{23}m$$

Solution

Question 2

$$32.64\ m$$

$$31.86\ m$$

$$52\ m$$

$$30.56\ m$$

Solution

Question 3

a. $$30\sqrt{2}$$b. $$60\hat{i} + 80\hat{j}$$c. $$30\hat{i} - 40\hat{j}$$

a. $$30\sqrt{2}$$b. $$60\hat{i} + 90\hat{j}$$c. $$30\hat{i} - 40\hat{j}$$

a. $$30\sqrt{2}$$b. $$60\hat{i} + 80\hat{j}$$c. $$30\hat{i} - 50\hat{j}$$

a. $$30\sqrt{2}$$b. $$60\hat{i} + 80\hat{j}$$c. $$40\hat{i} - 40\hat{j}$$

Solution

Question 4

$$=\dfrac{3}{2}\sqrt{\dfrac{5}{2}}ms^{-1}$$

$$=\dfrac{3}{22}\sqrt{\dfrac{5}{2}}ms^{-1}$$

$$=\dfrac{23}{2}\sqrt{\dfrac{5}{2}}ms^{-1}$$

$$=\dfrac{13}{2}\sqrt{\dfrac{5}{2}}ms^{-1}$$

Solution

Question 5

Bomb from $$B_1$$ reaches the ground first

Bomb from $$B_2$$ reaches the ground first

Bomb from $$B_3$$ reaches the ground first

They reach the ground simultaneously

Solution

Question 6

$$\nu = \dfrac{mu}{2m + M}$$

$$\nu = \dfrac{mu}{m + M}$$

$$\nu = \dfrac{2mu}{3m + M}$$

$$\nu = \dfrac{mu}{m + 4M}$$

Solution

Question 7

(i) - b, (ii) - a, (iii) - d, (iv) - d.

(i) - b, (ii) - a, (iii) - d, (iv) - c.

(i) - a, (ii) - b, (iii) - d, (iv) - c.

(i) - b, (ii) - c, (iii) - d, (iv) - a.

Solution

Question 8

9

8

10

11

Solution

Question 9

5

6

7

8

Solution

Question 10

33

3

4

5

Solution

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a. $$30\sqrt{2}$$
b. $$60\hat{i} + 80\hat{j}$$
c. $$30\hat{i} - 40\hat{j}$$

a. $$30\sqrt{2}$$
b. $$60\hat{i} + 90\hat{j}$$
c. $$30\hat{i} - 40\hat{j}$$

a. $$30\sqrt{2}$$
b. $$60\hat{i} + 80\hat{j}$$
c. $$30\hat{i} - 50\hat{j}$$

a. $$30\sqrt{2}$$
b. $$60\hat{i} + 80\hat{j}$$
c. $$40\hat{i} - 40\hat{j}$$