A projectile is thrown with velocity $v$ at an angle $\theta$ with horizontal. When the projectile is at a height equal to half of the maximum height, the vertical component of the velocity of projectile is ...........
A projectile is thrown with velocity $v$ at an angle $\theta$ with horizontal. When the projectile is at a height equal to half of the maximum height, the vertical component of the velocity of projectile is ...........
- A
$v \sin \theta \times 3$
- B
$\frac{v \sin \theta}{3}$
- C
$\frac{v \sin \theta}{\sqrt{2}}$
- D
$\frac{v \sin \theta}{\sqrt{3}}$
Similar Questions
A boy travelling in an open car moving on a levelled road with constant speed tosses a ball vertically up in the air and catches it back. Sketch the motion of the ball as observed by a boy standing on the footpath. Give explanation to support your diagram.
A boy travelling in an open car moving on a levelled road with constant speed tosses a ball vertically up in the air and catches it back. Sketch the motion of the ball as observed by a boy standing on the footpath. Give explanation to support your diagram.
Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in cartesian co-ordinates $A=A_{x} \hat{i}+A_{y} \hat{j},$ where $\hat{i}$ and $\hat{\jmath}$ are unit vector along $x$ and $y$ - directions, respectively and $A_{x}$ and $A_{y}$ are corresponding components of $A$. Motion can also be studied by expressing vectors in circular polar co-ordinates as $\overrightarrow A \, = \,{A_r}\widehat r\,\, + \,{A_\theta }\hat \theta $ where $\hat{r}=\frac{r}{r}=\cos \theta \hat{i}+\sin \theta \hat{\jmath}$ and $\hat{\theta}=-\sin \theta \hat{i}+\cos \theta \hat{j}$ are unit vectors along direction in which $\hat{r}$ and $\hat{\theta}$ are increasing.
$(a)$ Express ${\widehat {i\,}}$ and ${\widehat {j\,}}$ in terms of ${\widehat {r\,}}$ and ${\widehat {\theta }}$ .
$(b)$ Show that both $\widehat r$ and $\widehat \theta $ are unit vectors and are perpendicular to each other.
$(c)$ Show that $\frac{d}{{dr}}(\widehat r)\, = \,\omega \hat \theta \,$, where $\omega \, = \,\frac{{d\theta }}{{dt}}$ and $\frac{d}{{dt}}(\widehat \theta )\, = \, - \theta \widehat r\,$.
$(d)$ For a particle moving along a spiral given by $\overrightarrow r \, = \,a\theta \widehat r$, where $a = 1$ (unit), find dimensions of $a$.
$(e)$ Find velocity and acceleration in polar vector representation for particle moving along spiral described in $(d)$ above.
Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in cartesian co-ordinates $A=A_{x} \hat{i}+A_{y} \hat{j},$ where $\hat{i}$ and $\hat{\jmath}$ are unit vector along $x$ and $y$ - directions, respectively and $A_{x}$ and $A_{y}$ are corresponding components of $A$. Motion can also be studied by expressing vectors in circular polar co-ordinates as $\overrightarrow A \, = \,{A_r}\widehat r\,\, + \,{A_\theta }\hat \theta $ where $\hat{r}=\frac{r}{r}=\cos \theta \hat{i}+\sin \theta \hat{\jmath}$ and $\hat{\theta}=-\sin \theta \hat{i}+\cos \theta \hat{j}$ are unit vectors along direction in which $\hat{r}$ and $\hat{\theta}$ are increasing.
$(a)$ Express ${\widehat {i\,}}$ and ${\widehat {j\,}}$ in terms of ${\widehat {r\,}}$ and ${\widehat {\theta }}$ .
$(b)$ Show that both $\widehat r$ and $\widehat \theta $ are unit vectors and are perpendicular to each other.
$(c)$ Show that $\frac{d}{{dr}}(\widehat r)\, = \,\omega \hat \theta \,$, where $\omega \, = \,\frac{{d\theta }}{{dt}}$ and $\frac{d}{{dt}}(\widehat \theta )\, = \, - \theta \widehat r\,$.
$(d)$ For a particle moving along a spiral given by $\overrightarrow r \, = \,a\theta \widehat r$, where $a = 1$ (unit), find dimensions of $a$.
$(e)$ Find velocity and acceleration in polar vector representation for particle moving along spiral described in $(d)$ above.
A projectile is launched from the origin in the $xy$ plane ( $x$ is the horizontal and $y$ is the vertically up direction) making an angle $\alpha$ from the $x$-axis. If its distance. $r =\sqrt{ x ^2+ y ^2}$ from the origin is plotted against $x$, the resulting curves show different behaviours for launch angles $\alpha_1$ and $\alpha_2$ as shown in the figure below. For $\alpha_1, r ( x )$ keeps increasing with $x$ while for $\alpha_2$, $r(x)$ increases and reaches a maximum, then decreases and goes through a minimum before increasing again. The switch between these two cases takes place at an angle $\alpha_c\left(\alpha_1 < \alpha_c < \alpha_2\right)$. The value of $\alpha_c$ is [ignore where $v_0$ is the initial speed of the projectile and $g$ is the acceleration due to gravity]
A projectile is launched from the origin in the $xy$ plane ( $x$ is the horizontal and $y$ is the vertically up direction) making an angle $\alpha$ from the $x$-axis. If its distance. $r =\sqrt{ x ^2+ y ^2}$ from the origin is plotted against $x$, the resulting curves show different behaviours for launch angles $\alpha_1$ and $\alpha_2$ as shown in the figure below. For $\alpha_1, r ( x )$ keeps increasing with $x$ while for $\alpha_2$, $r(x)$ increases and reaches a maximum, then decreases and goes through a minimum before increasing again. The switch between these two cases takes place at an angle $\alpha_c\left(\alpha_1 < \alpha_c < \alpha_2\right)$. The value of $\alpha_c$ is [ignore where $v_0$ is the initial speed of the projectile and $g$ is the acceleration due to gravity]
- [KVPY 2021]
A boy throws a ball in air at $60^o$ to the horizontal, along a road with a speed of $10\,ms^{-1}$ $(36\, km/h)$. Another boy sitting in a passing by car observes the ball. Sketch the motion of the ball as observed by the boy in the car, if car has a speed of $(18\, km/h)$. Give explanation to support your diagram.
A boy throws a ball in air at $60^o$ to the horizontal, along a road with a speed of $10\,ms^{-1}$ $(36\, km/h)$. Another boy sitting in a passing by car observes the ball. Sketch the motion of the ball as observed by the boy in the car, if car has a speed of $(18\, km/h)$. Give explanation to support your diagram.
A projectile is thrown at an angle $\theta$ such that it is just able to cross a vertical wall at its highest point as shown in the figure.The angle $\theta$ at which the projectile is thrown is given by
A projectile is thrown at an angle $\theta$ such that it is just able to cross a vertical wall at its highest point as shown in the figure.The angle $\theta$ at which the projectile is thrown is given by