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(A) The slope $(m)$ of the trajectory at any time $(t)$ is given by the tangent of the angle $\phi$ that the velocity vector makes with the horizontal

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(A) The slope $(m)$ of the trajectory at any time $(t)$ is given by the tangent of the angle $\phi$ that the velocity vector makes with the horizontal.The velocity components at time $(t)$ are $v_x = u \cos 	heta$ and $v_y = u \sin 	heta - g t$.The slope $(m)$ is given by:$m = 	an \phi = \frac{v_y}{v_x} = \frac{u \sin 	heta - g t}{u \cos 	heta}$Simplifying this expression:$m = \frac{u \sin 	heta}{u \cos 	heta} - \frac{g t}{u \cos 	heta}$$m = 	an 	heta - \left( \frac{g}{u \cos 	heta} ight) t$This equation is of the form $m = c - k t$,where $c = 	an 	heta$ (a constant) and $k = \frac{g}{u \cos 	heta}$ (a positive constant).This represents a straight line with a positive $y$-intercept $(	an 	heta)$ and a negative slope $(-k)$. Therefore,the graph of $m$ versus $t$ is a straight line with a negative slope.

$A$ particle is projected at an angle $\theta$ with the horizontal from the ground. The slope $(m)$ of the trajectory of the particle varies with time $(t)$ as:

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