Name: Motion on an Inclined Plane
Uploaded: 2019-07-19
Duration: 6 min 34 s
Description: We all are curious about gravity. Through this video, let us learn to break the force of gravity into components in case of motion on an inclined plane.

Question

Show that the horizontal range of a projectile with initial speed

u

and angle of projection

θ

is

\frac{u^{2} \sin 2 θ}{g}

. At what angle of projection is the horizontal range maximum for a given initial speed of the projectile? What is this maximum range?

Accepted Answer

When a body is thrown upward in a direction different from the vertical, then it is acted upon by acceleration due to gravity (which acts vertically downwards) and moves along a curved path in a vertical plane. This motion is called the 'projectile motion' and its path is called the 'trajectory'.

Suppose a body is projected up with an initial velocity $u$ in a direction making an angle $θ_{0}$ with the horizontal. Resolving $u$ in horizontal and vertical components, we have,
$u_{x} = ucosθ$ and

$u_{y} = usinθ$
The body moves under constant gravitational acceleration $g$ acting vertically downwards. Therefore, the horizontal velocity remains unchanged throughout the motion (provided the resistance of air is negligible). But the vertical velocity continuously changes.

The body covers a horizontal distance $O O^{'}$ during its flight. This distance is called the ‘horizontal range' $R$ . Its value is given by,

$R = u_{x} \times T$ $= (ucosθ) \times \frac{2 usinθ}{g} = u^{2} \frac{(2 sinθcosθ)}{g}$

Thus, $R = \frac{u^{2} \sin 2 θ}{g}$

Clearly, for a maximum horizontal range, $2 θ = 1$ or $2 θ = 90 °$ or $θ = 45 °$ .

The maximum range will be, $R_{\max} = \frac{u^{2}}{g}$

Show that the horizontal range of a projectile with initial speed $u$ and angle of projection $θ$ is $\frac{u^{2} \sin 2 θ}{g}$ . At what angle of projection is the horizontal range maximum for a given initial speed of the projectile? What is this maximum range?

Important Questions on Motion in One and Two Dimensions

Learn and Prepare for any exam you want !

Show that the horizontal range of a projectile with initial speed u and angle of projection θ is u2sin2θg . At what angle of projection is the horizontal range maximum for a given initial speed of the projectile? What is this maximum range?

Important Questions on Motion in One and Two Dimensions

Learn and Prepare for any exam you want !

Show that the horizontal range of a projectile with initial speed $u$ and angle of projection $θ$ is $\frac{u^{2} \sin 2 θ}{g}$ . At what angle of projection is the horizontal range maximum for a given initial speed of the projectile? What is this maximum range?