Motion in a Plane Chapter-Wise Test 8

Correct answer Carries: 4.

Wrong Answer Carries: -1.

An object moves in a circle of radius 2 m with a constant speed of 4 m/s. What is its angular speed?

Angular speed ω = v / R.

Given: v = 4 m/s, R = 2 m.

ω = 4 / 2 = 2 rad/s.

1 rad/s
2 rad/s
4 rad/s
8 rad/s
2

A ball is thrown at \( 20 \, \text{m/s} \) at \( 45^\circ \) from a height of \( 5 \, \text{m} \). What is its range? (Take \( g = 10 \, \text{m/s}^2 \))

Time of flight: \( y = v_{0y} t - \frac{1}{2} g t^2 \), where \( y = -5 \, \text{m}, v_{0y} = 20 \sin 45^\circ = 10\sqrt{2} \).

\( -5 = 10\sqrt{2} t - 5 t^2 \Rightarrow 5 t^2 - 10\sqrt{2} t - 5 = 0 \Rightarrow t^2 - 2\sqrt{2} t - 1 = 0 \).

Solve: \( t = \frac{2\sqrt{2} \pm \sqrt{8 + 4}}{2} = \frac{2\sqrt{2} + \sqrt{12}}{2} = \sqrt{2} + \sqrt{3} \approx 3.146 \, \text{s} \).

Range \( R = v_{0x} t = 20 \cos 45^\circ \times 3.146 = 10\sqrt{2} \times 3.146 \approx 44.41 \, \text{m} \).

40 m
50 m
30 m
44.41 m
4

A particle starts with velocity \( 5 \hat{i} \, \text{m/s} \) and accelerates at \( (-3 \hat{j}) \, \text{m/s}^2 \). What is its displacement magnitude after \( 2 \, \text{s} \)?

Displacement \( \mathbf{r} = \mathbf{v}_0 t + \frac{1}{2} \mathbf{a} t^2 \).

\( x = v_{0x} t = 5 \times 2 = 10 \, \text{m} \).

\( y = \frac{1}{2} a_y t^2 = \frac{1}{2} \times (-3) \times 2^2 = -6 \, \text{m} \).

Magnitude \( |\mathbf{r}| = \sqrt{x^2 + y^2} = \sqrt{10^2 + (-6)^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.66 \, \text{m} \).

10 m
15 m
11.66 m
8 m
3

An object in uniform circular motion has a centripetal acceleration of 16 m/s² and a radius of 4 m. What is its linear speed?

Centripetal acceleration a_c = v² / R.

Given: a_c = 16 m/s², R = 4 m.

16 = v² / 4 ⇒ v² = 64 ⇒ v = √64 = 8 m/s.

4 m/s
16 m/s
8 m/s
12 m/s
3

A car moves north at \( 12 \, \text{m/s} \) while a wind blows east at \( 5 \, \text{m/s} \). What is the magnitude of the car’s velocity relative to the ground?

Velocity components: \( v_x = 5 \, \text{m/s}, v_y = 12 \, \text{m/s} \).

Magnitude \( v = \sqrt{v_x^2 + v_y^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{m/s} \).

12 m/s
13 m/s
17 m/s
10 m/s
2

A particle moves in a circle of radius \( 5 \, \text{m} \) with an angular speed of \( 2 \, \text{rad/s} \). What is its centripetal acceleration?

Centripetal acceleration \( a_c = \omega^2 R \).

Given: \( \omega = 2 \, \text{rad/s}, R = 5 \, \text{m} \).

\( a_c = 2^2 \times 5 = 4 \times 5 = 20 \, \text{m/s}^2 \).

10 m/s²
20 m/s²
25 m/s²
15 m/s²
2

A ball is dropped from a height of \( 44.1 \, \text{m} \) with a horizontal speed of \( 7 \, \text{m/s} \). What is its speed on hitting the ground? (Take \( g = 9.8 \, \text{m/s}^2 \))

Vertical velocity: \( v_y = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 44.1} = \sqrt{863.28} \approx 29.38 \, \text{m/s} \).

Horizontal velocity: \( v_x = 7 \, \text{m/s} \).

Speed \( v = \sqrt{v_x^2 + v_y^2} = \sqrt{7^2 + 29.38^2} = \sqrt{49 + 863.28} = \sqrt{912.28} \approx 30.2 \, \text{m/s} \).

25 m/s
30.2 m/s
35 m/s
20 m/s
2

A particle in circular motion has a centripetal acceleration of \( 36 \, \text{m/s}^2 \) and radius \( 2 \, \text{m} \). What is its speed?

Centripetal acceleration \( a_c = \frac{v^2}{R} \).

Given: \( a_c = 36 \, \text{m/s}^2, R = 2 \, \text{m} \).

\( 36 = \frac{v^2}{2} \Rightarrow v^2 = 72 \Rightarrow v = \sqrt{72} = 6\sqrt{2} \approx 8.49 \, \text{m/s} \).

6 m/s
10 m/s
8.49 m/s
12 m/s
3

What is the effect on the range of a projectile if the launch angle is increased beyond 45° on level ground?

The range \( R = \frac{v_0^2 \sin 2\theta_0}{g} \) is maximized at \( \theta_0 = 45^\circ \) (when \( 2\theta_0 = 90^\circ \)). Beyond 45°, \( \sin 2\theta_0 \) decreases (e.g., at 60°, \( 2\theta_0 = 120^\circ \), \( \sin 120^\circ < 1 \)), reducing the range.

It increases continuously
It decreases
It remains the same
It becomes zero
2

A cyclist moves east at \( 18 \, \text{m/s} \) while the wind blows south at \( 24 \, \text{m/s} \). What is the magnitude of the relative velocity of the wind with respect to the cyclist?

Relative velocity \( \mathbf{v}_{wc} = \mathbf{v}_w - \mathbf{v}_c \), where \( \mathbf{v}_c = 18 \hat{i}, \mathbf{v}_w = -24 \hat{j} \).

\( \mathbf{v}_{wc} = -24 \hat{j} - 18 \hat{i} = -18 \hat{i} - 24 \hat{j} \).

Magnitude \( |\mathbf{v}_{wc}| = \sqrt{(-18)^2 + (-24)^2} = \sqrt{324 + 576} = \sqrt{900} = 30 \, \text{m/s} \).

20 m/s
30 m/s
42 m/s
25 m/s
2

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