Motion in a Plane Chapter-Wise Test 15

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A particle in circular motion has a centripetal acceleration of \( 25 \, \text{m/s}^2 \) and a speed of \( 10 \, \text{m/s} \). What is the radius of the path?

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

Given: \( a_c = 25 \, \text{m/s}^2, v = 10 \, \text{m/s} \).

\( 25 = \frac{10^2}{R} \Rightarrow 25 = \frac{100}{R} \Rightarrow R = \frac{100}{25} = 4 \, \text{m} \).

2 m
8 m
6 m
4 m
4

A projectile is launched with a speed of 40 m/s at an angle of 30°. What is its velocity at the highest point? (Take g = 10 m/s²)

At the highest point, v_y = 0, v_x = v₀ cos θ₀.

Given: v₀ = 40 m/s, cos 30° = √3/2.

v_x = 40 × (√3/2) = 20 √3 ≈ 34.64 m/s.

20 m/s
40 m/s
34.64 m/s
0 m/s
3

A particle moves in a circular path with a period of \( 4 \, \text{s} \) and radius \( 6 \, \text{m} \). What is its linear speed?

Angular speed \( \omega = \frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2} \, \text{rad/s} \).

Linear speed \( v = \omega R = \frac{\pi}{2} \times 6 = 3\pi \approx 9.42 \, \text{m/s} \).

6 m/s
9.42 m/s
12 m/s
3 m/s
2

A ball is thrown horizontally at \( 12 \, \text{m/s} \) from a height of \( 19.6 \, \text{m} \). What is its speed on hitting the ground? (Take \( g = 9.8 \, \text{m/s}^2 \))

Vertical velocity: \( v_y = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 19.6} = \sqrt{384.16} \approx 19.6 \, \text{m/s} \).

Horizontal velocity: \( v_x = 12 \, \text{m/s} \).

Speed \( v = \sqrt{v_x^2 + v_y^2} = \sqrt{12^2 + 19.6^2} = \sqrt{144 + 384.16} = \sqrt{528.16} \approx 22.98 \, \text{m/s} \).

20 m/s
22.98 m/s
25 m/s
15 m/s
2

In uniform circular motion, what is the relationship between the velocity vector and the path of the object?

In uniform circular motion, the velocity vector is always tangential to the circular path, directed along the instantaneous direction of motion, perpendicular to the radius at every point.

It is tangential to the path
It is radial towards the center
It is radial away from the center
It is perpendicular to the plane of motion
1

A projectile is launched at \( 25 \, \text{m/s} \) at \( 37^\circ \). What is its vertical velocity at \( t = 1 \, \text{s} \)? (Take \( g = 10 \, \text{m/s}^2, \sin 37^\circ = 0.6 \))

Vertical velocity \( v_y = v_0 \sin \theta_0 - g t \).

Given: \( v_0 = 25 \, \text{m/s}, \sin 37^\circ = 0.6, t = 1 \, \text{s}, g = 10 \, \text{m/s}^2 \).

\( v_y = 25 \times 0.6 - 10 \times 1 = 15 - 10 = 5 \, \text{m/s} \).

10 m/s
15 m/s
5 m/s
0 m/s
3

In motion with constant acceleration in a plane, what is true about the acceleration vector?

The acceleration vector in motion with constant acceleration remains constant in both magnitude and direction. This allows the motion to be analyzed as two independent one-dimensional motions along perpendicular axes.

It changes direction
It changes magnitude
It varies with velocity
It is constant in magnitude and direction
2

A projectile is launched at \( 34 \, \text{m/s} \) at \( 60^\circ \). What is its horizontal range? (Take \( g = 9.8 \, \text{m/s}^2, \sin 120^\circ = 0.866 \))

Range \( R = \frac{v_0^2 \sin 2\theta_0}{g} \), where \( 2\theta_0 = 120^\circ \).

Given: \( v_0 = 34 \, \text{m/s}, \sin 120^\circ = 0.866, g = 9.8 \, \text{m/s}^2 \).

\( R = \frac{34^2 \times 0.866}{9.8} = \frac{1156 \times 0.866}{9.8} \approx \frac{1001.096}{9.8} \approx 102.15 \, \text{m} \).

102.15 m
50 m
120 m
80 m
1

A particle has an initial velocity of 3 î m/s and a constant acceleration of 2 ĵ m/s². What is the magnitude of its displacement after 3 s?

Displacement r = v₀ t + (1/2) a t².

x = v₀x t = 3 × 3 = 9 m.

y = (1/2) a_y t² = (1/2) × 2 × 3² = 9 m.

Magnitude |r| = √(x² + y²) = √(9² + 9²) = √162 ≈ 12.73 m.

9 m
18 m
12.73 m
15 m
3

A projectile is launched at \( 26 \, \text{m/s} \) at \( 53^\circ \). What is its horizontal velocity component? (Take \( \cos 53^\circ = 0.6 \))

Horizontal velocity \( v_x = v_0 \cos \theta_0 \).

Given: \( v_0 = 26 \, \text{m/s}, \cos 53^\circ = 0.6 \).

\( v_x = 26 \times 0.6 = 15.6 \, \text{m/s} \).

12 m/s
20 m/s
15.6 m/s
10 m/s
3

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