First slide

Projectile motion

Question

A projectile is given an initial velocity of $\hat{i} + 2 \hat{j}$ . The Cartesian equation of its path is: $(g = 10 m / s^{2})$

Moderate

A

$y = 2 x - 5 x^{2}$
B

$y = x - 5 x^{2}$
C

$4 y = 2 x - 5 x^{2}$
D

$y = 2 x - 25 x^{2}$

Solution

$\vec{u} = u_{x} \hat{i} + u_{y} \hat{j} = u \cos θ \hat{i} + u \sin θ \hat{j} = \hat{i} + 2 \hat{j}$
$∴$ $\tan θ = \frac{u \sin θ}{u \cos θ} = \frac{2}{1}$ , $\cos θ = \frac{1}{\sqrt{5}}$
The desired equation is, $y = x \tan θ - \frac{g x^{2}}{2 u^{2} \cos^{2} θ}$
$= x \times 2 - \frac{10 x^{2}}{2 {(\sqrt{2^{2} + 1^{2}})}^{2} {(\frac{1}{\sqrt{5}})}^{2}}$
or $y = 2 x - 5 x^{2}$

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