A stone is projected from the ground with a velocity of 14 m/s. One second later, it clears a wall 2 m high. What is the angle of projection?
Understand the Problem
The question is asking to determine the angle of projection for a stone that is thrown with a certain initial velocity and must clear a wall of specified height after one second. This involves applying principles from physics related to projectile motion.
Answer
$30^\circ$
Answer for screen readers
The angle of projection is $30^\circ$.
Steps to Solve
- Identify initial parameters
The stone is projected with an initial velocity $v_0 = 14 \text{ m/s}$, and it must clear a wall of height $h = 2 \text{ m}$ after $t = 1 \text{ s}$. The acceleration due to gravity is $g = 10 \text{ m/s}^2$.
- Use the vertical motion equation
The height of the stone after time $t$ can be calculated using the equation for vertical motion:
$$ h = v_{0y} t - \frac{1}{2} g t^2 $$
where $v_{0y} = v_0 \sin(\theta)$ is the vertical component of the initial velocity.
- Set up the equation
Plug in the values into the height equation:
$$ 2 = (14 \sin(\theta))(1) - \frac{1}{2}(10)(1^2) $$
This simplifies to:
$$ 2 = 14 \sin(\theta) - 5 $$
- Simplify the equation
Rearranging the above equation gives:
$$ 14 \sin(\theta) = 2 + 5 $$
$$ 14 \sin(\theta) = 7 $$
- Solve for $\sin(\theta)$
Divide both sides by 14:
$$ \sin(\theta) = \frac{7}{14} = \frac{1}{2} $$
- Determine the angle of projection
The angle whose sine is $\frac{1}{2}$ is:
$$ \theta = 30^\circ $$
The angle of projection is $30^\circ$.
More Information
In projectile motion, the angle of projection directly influences the height and distance covered. The angle $30^\circ$ is significant because it yields the necessary vertical displacement to clear the wall at the specified height.
Tips
- Confusing vertical and horizontal components: Ensure to separately consider vertical motion using $v_{0y} = v_0 \sin(\theta)$.
- Neglecting gravity's effect: Always remember to include the term $-\frac{1}{2} g t^2$ in height calculations.
AI-generated content may contain errors. Please verify critical information
