Question

A body is thrown with a velocity \( (5\hat{i} + 6\hat{j}) \, \text{m/s} \). Its maximum height is (given that acceleration due to gravity = \( 10 \, \text{m/s}^2 \)):

• A. \(12.5 \, {m}\)
• B. \(1.8 \, {m}\)
• C. \(1.25 \, {m}\)
• D. \(0.9 \, {m}\)

Hint:

In projectile motion, only the vertical component of the initial velocity affects the maximum height reached. Remember, gravity only acts in the vertical direction and affects the vertical motion.

Answer 1

The Correct Option is B

To find the maximum height, we'll consider the vertical motion of the body.

1. Vertical component of velocity: The initial vertical velocity (\( u_y \)) is given by the \( j \)-component of the velocity vector:

\[ u_y = 6 \text{ m/s} \]

2. Kinematic equation: We can use the following kinematic equation to find the maximum height (\( H \)):

\[ v_y^2 = u_y^2 + 2as \]

where:

• \( v_y = 0 \) m/s (final vertical velocity at maximum height)
• \( u_y = 6 \) m/s (initial vertical velocity)
• \( a = -10 \) m/s\(^2\) (acceleration due to gravity, negative since it acts downwards)
• \( s = H \) (displacement, which is the maximum height)

3. Substitute and solve for \( H \):

\[ 0^2 = 6^2 + 2 \times (-10) \times H \] \[ 0 = 36 - 20H \] \[ 20H = 36 \] \[ H = \frac{36}{20} = 1.8 \text{ m} \]

Therefore, the maximum height reached by the body is 1.8 m.

Answer 2

To solve this problem, we need to calculate the maximum height reached by a body thrown with the given velocity in the problem. The velocity is given as a vector, and the acceleration due to gravity is provided as 10 m/s².

1. Breaking the Velocity into Components:
The velocity of the body is given as \( (5i + 6j) \, \text{m/s} \). Here, \( i \) and \( j \) are unit vectors in the x and y directions respectively. The maximum height is determined by the vertical component of the velocity.

The vertical component of the velocity \( v_y = 6 \, \text{m/s} \), as it is the coefficient of the \( j \)-component.

2. Using the Equation for Maximum Height:
We can use the following kinematic equation to find the maximum height:

\[ v_y^2 = u_y^2 - 2g h \] where: - \( v_y = 0 \, \text{m/s} \) (at maximum height, the vertical velocity becomes zero), - \( u_y = 6 \, \text{m/s} \) (initial vertical velocity), - \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity), - \( h \) is the maximum height. Thus, the equation becomes: \[ 0 = (6)^2 - 2 \times 10 \times h \] \[ 36 = 20h \] \[ h = \frac{36}{20} = 1.8 \, \text{m} \]

3. Conclusion:
The maximum height reached by the body is 1.8 m.

Final Answer:
The correct answer is Option B: 1.8 m.