Name: Rectilinear Motion and Its Example
Uploaded: 2019-07-19
Duration: 11 min 6 s
Description: What is the similarity between the motion of people marching and the motion of a ball rolling down an inclined plane? Let's get an answer in this video.

Question

Derive the formula for the range and maximum height achieved by a projectile thrown from the origin with initial velocity

\vec{u}

at an angel

θ

to the horizontal.

Accepted Answer

the components of velocity at time t are given by

$v_{x} = u_{x} = u \cos θ$ ......(1)

$v_{y} = u_{y} - g t = u \sin θ - g t$ .....(2)

as,

$s_{x} = ucosθ . t . . . . . . (3) s_{y} = usinθ . t - \frac{1}{2} {gt}^{2} . . . . . . . (4)$

The direction of motion of the projectile at any time $t$ makes an angle $α$ with the horizontal which is given by

$tanα = \frac{v_{y} (t)}{v_{x} (t)}$

Let the time taken by the projectile to reach the maximum height be $t_{0}$ .

at $t_{0}$ , $v_{y} = 0$

using eq (2)

$0 = u \sin θ - g t_{0} t_{0} = \frac{u \sin θ}{g}$

. ·. Total time in air $= T = 2 t_{0}$ is the time of flight.

The total horizontal distance travelled by the particle in this time $T$ can be obtained by using Eq. (3)

Range $= R = u_{x}$

$T = ucosθ . 2 t_{0} = ucosθ . \frac{2 usinθ}{g}$

$= \frac{u^{2} (2 sinθcosθ)}{g} = \frac{u^{2} \sin 2 θ}{g}$

for maximum range $θ = 45 °$

maximum height can be calculated from eq(4)

$s_{y} = u \sin θ . t_{0} - \frac{1}{2} g t_{0}^{2} = u \sin θ . \frac{u \sin θ}{g} - \frac{1}{2} g {(\frac{u \sin θ}{g})}^{2} = \frac{u^{2} \sin^{2} θ}{2 g} = \frac{u_{y}^{2}}{2 g}$

Derive the formula for the range and maximum height achieved by a projectile thrown from the origin with initial velocity $\vec{u}$ at an angel $θ$ to the horizontal.

Important Questions on Motion in a Plane

Learn and Prepare for any exam you want !

Derive the formula for the range and maximum height achieved by a projectile thrown from the origin with initial velocity u→ at an angel θ to the horizontal.

Important Questions on Motion in a Plane

Learn and Prepare for any exam you want !

Derive the formula for the range and maximum height achieved by a projectile thrown from the origin with initial velocity $\vec{u}$ at an angel $θ$ to the horizontal.