Question

A particle at rest starts from the origin with a constant acceleration a that makes an angle 60° with the positive y-axis. If its displacement along y-axis is 10m in time 2s,then the magnitude of a is:

• A. 10ms^-2
• B. 4ms^-2
• C. 8ms^-2
• D. 15ms^-2
• E. 20ms^-2

Answer 1

The Correct Option is A

Given:

• Particle starts from rest at the origin.
• Constant acceleration \( \vec{a} \) makes a \( 60^\circ \) angle with the positive y-axis.
• Displacement along y-axis is 10 m in 2 seconds.

Step 1: Resolve Acceleration into Components

The acceleration \( \vec{a} \) can be resolved into:

\[ a_y = a \cos 60^\circ = \frac{a}{2} \]

\[ a_x = a \sin 60^\circ = \frac{a\sqrt{3}}{2} \]

Step 2: Apply Kinematic Equation for y-Displacement

Since the particle starts from rest, its initial velocity \( u_y = 0 \). The displacement along the y-axis is given by:

\[ y = u_y t + \frac{1}{2} a_y t^2 \]

Substitute \( y = 10 \, \text{m} \), \( t = 2 \, \text{s} \), and \( a_y = \frac{a}{2} \):

\[ 10 = 0 + \frac{1}{2} \left( \frac{a}{2} \right) (2)^2 \]

\[ 10 = \frac{a}{4} \times 4 \]

\[ 10 = a \]

Conclusion:

The magnitude of the acceleration \( \vec{a} \) is 10 ms\(^{-2}\).

Answer: \(\boxed{A}\)

Answer 2

Step 1: Decompose the Acceleration Vector

The acceleration \( \vec{a} \) can be expressed in terms of its components along the x- and y-axes:

\[ a_y = a \cos 60^\circ = \frac{a}{2}, \quad a_x = a \sin 60^\circ = \frac{a\sqrt{3}}{2}. \]

Here, \( a_y \) represents the component of acceleration along the y-axis, which governs the motion in the vertical direction.

Step 2: Use Kinematic Equation for y-Displacement

The particle starts from rest (\( u_y = 0 \)), so its displacement along the y-axis after \( t = 2 \, \text{s} \) is governed by the kinematic equation:

\[ y = u_y t + \frac{1}{2} a_y t^2. \]

Substituting the known values (\( y = 10 \, \text{m} \), \( t = 2 \, \text{s} \), and \( a_y = \frac{a}{2} \)):

\[ 10 = 0 + \frac{1}{2} \left( \frac{a}{2} \right) (2)^2. \]

Simplify the equation:

\[ 10 = \frac{1}{2} \cdot \frac{a}{2} \cdot 4, \]

\[ 10 = \frac{a}{4} \cdot 4, \]

\[ 10 = a. \]

Conclusion:

The magnitude of the acceleration \( \vec{a} \) is \( \boxed{10 \, \text{ms}^{-2}} \).