Projectile motion

Two projectiles thrown at 30° and 45° with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is :

If the initial velocity in horizontal direction of a projectile is unit vector $\hat i$ and the equation of trajectory is $y=5x(1-x)$. The $y$ component vector of the initial velocity is ______ $\hat j$.

(Take $g=10$ m/s$^2$)

Two projectiles are thrown with same initial velocity making an angle of 45° and 30° with the horizontal respectively. The ratio of their respective ranges will be :

A ball of mass m is thrown vertically upward. Another ball of mass 2 m is thrown at an angle θ with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is $\dfrac1x$. The value of $x$ is ______.

A body of mass 10 kg is projected at an angle of 45° with the horizontal. The trajectory of the body is observed to pass through a point (20, 10). If T is the time of flight, then its momentum vector, at time $t=\dfrac T{\sqrt2}$, is ______.

[Take g = 10 m/s$^2$]

A uniform electric field E = (8m/e) V/m is created between two parallel plates of length 1 m as shown in figure, (where m = mass of electron and e = charge of electron). An electron enters the field symmetrically between the plates with a speed of 2 m/s. The angle of the deviation (θ) of the path of the electron as it comes out of the field will be ______.

A ball is projected with kinetic energy E, at an angle of 60° to the horizontal. The kinetic energy of this ball at the highest point of its flight will become :

An object is projected in the air with initial velocity u at an angle θ. The projectile motion is such that the horizontal range R, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be _______ degree.

Two projectiles are thrown towards each other at 15° and 45° angles respectively with the horizontal at the same speed. The difference between the horizontal distances traveled by the two projectiles is 80 m. The initial speed of the projectiles is _______ m s$^{-1}$ [Given g = 10 m s$^{-2}$]

The value of $t$ is ____.

The value of $x$ is ____.

A uniform electric field, $\vec E = -400\sqrt3\,\hat y\ \mathrm{NC^{-1}}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of $2\sqrt{10}\times10^{6}\ \mathrm{m\,s^{-1}}$. This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure. Take $\dfrac qm = 10^{10}\ \mathrm{C\,kg^{-1}}$. Then

A ball is thrown from ground at an angle $\theta$ with horizontal and with an initial speed $u_0$. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $V_1$. After hitting the ground, the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_0/\alpha$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8V_1$, the value of $\alpha$ is ______.

A ball is projected from the ground at an angle of $45^\circ$ with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $30^\circ$ with the horizontal surface. The maximum height it reaches after the bounce, in metres, is ___________.

A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance of 1.2 m from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the list.

The position of a projectile launched from the origin at $t=0$ is given by $\vec r=\left(40\hat i+50\hat j\right)$ m at $t=2$ s. If the projectile was launched at an angle $\theta$ from the horizontal, then $\theta$ is (take $g=10$ ms$^{-2}$).

The initial speed of a bullet fired from a rifle is 630 m/s. The rifle is fired at the centre of a target 700 m away at the same level as the target. How far above the centre of the target the rifle must be aimed in order to hit the target ?

A ball of mass 160 g is thrown up at an angle of 60° to the horizontal at a speed of 10 ms$^{-1}$. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly ($g=10$ ms$^{-2}$)

A particle of mass $m$ is projected from the ground with an initial speed $u_0$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_0$. The angle that the composite system makes with the horizontal immediately after the collision is

A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at $t=0$. It then executes simple harmonic motion with angular frequency $\omega=\dfrac\pi3$ rad/s. Simultaneously at $t=0$, a small pebble is projected with speed $v$ from point $P$ at an angle of 45° as shown in the figure. Point $P$ is at a horizontal distance of 10 m from $O$. If the pebble hits the block at $t=1$ s, the value of $v$ is (take $g=10\ \mathrm{m/s^2}$)