Question

The equation of trajectory of a projectile is given by \( y = x - 10x^2 \). Its speed of projection is – \((g = 10 \, \text{m/s}^2)\)

• A. \( 1 \, \text{m/s} \)
• B. \( 2 \, \text{m/s} \)
• C. \( 3 \, \text{m/s} \)
• D. \( 4 \, \text{m/s} \)

Hint:

Compare the standard projectile equation with the given equation to extract projection speed. Watch units and angle values like \( \theta = 45^\circ \).

Answer 1

The Correct Option is D

The general trajectory equation for a projectile is: \[ y = x \tan \theta - \frac{g}{2u^2 \cos^2 \theta} x^2 \] Comparing this with the given equation \( y = x - 10x^2 \), we get: \[ \tan \theta = 1 \Rightarrow \theta = 45^\circ \] \[ \frac{g}{2u^2 \cos^2 \theta} = 10 \] Since \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), we substitute: \[ \frac{10}{2u^2 \cdot \frac{1}{2}} = 10 \Rightarrow \frac{10}{u^2} = 10 \Rightarrow u^2 = 1 \Rightarrow u = \sqrt{1} = 1 \, \text{m/s} \] However, a mistake is evident here.
Let's resolve properly.
Given: \[ \frac{g}{2u^2 \cos^2\theta} = 10, \quad \tan\theta = 1 \Rightarrow \theta = 45^\circ \Rightarrow \cos^2\theta = \frac{1}{2} \] \[ \frac{10}{2u^2 \cdot \frac{1}{2}} = 10 \Rightarrow \frac{10}{u^2} = 10 \Rightarrow u^2 = 1 \Rightarrow u = 1 \] Wait, this again gives \( u = 1 \, \text{m/s} \), which contradicts the correct option being 4 m/s.
Let us double-check with correct substitution.
\[ \text{Given: } \frac{g}{2u^2 \cos^2\theta} = 10, \quad \cos^2(45^\circ) = \frac{1}{2}, \quad g = 10 \Rightarrow \frac{10}{2u^2 \cdot \frac{1}{2}} = 10 \Rightarrow \frac{10}{u^2} = 10 \Rightarrow u^2 = 1 \Rightarrow u = 1 \] So our initial assumption must be wrong — either the comparison is incorrect or actual coefficient is different.
Let's do direct coefficient matching: \[ y = x - 10x^2 \Rightarrow \text{coefficient of } x^2 = \frac{g}{2u^2 \cos^2\theta} \Rightarrow 10 = \frac{10}{2u^2 \cdot \frac{1}{2}} = \frac{10}{u^2} \Rightarrow u^2 = 1 \Rightarrow u = 1 \, \text{m/s} \] So the actual correct answer is (A) \( 1 \, \text{m/s} \).
The image answer key may be incorrect.