Comparing Two Projectiles

Two objects are projected with same velocity 'u' at different angles $\alpha$ and $\beta$ with the horizontal. If $\alpha + \beta = 90^\circ$, the ratio of horizontal range of the first object to the second object will be $[\sin(\pi - \theta) = \sin\theta]$

The two projectiles are projected with the same initial velocities at the $15^\circ$ and $30^\circ$ with respect to the horizontal. The ratio of their ranges is $1 : x$. The value of $x$ is

Two identical bodies, projected with the same speed at two different angles cover the same horizontal range $R$. If the time of flight of these bodies are 5 s and 10 s, respectively, then the value of $R$ is

______ m. (Take $g = 10\text{ m/s}^{2}$)

If two spheres are projected at angles $45^\circ$ and $60^\circ$ with horizontal and the maximum heights reached are same. The ratio of their initial velocities is ($\sin 45^\circ = \frac{1}{\sqrt{2}}, \sin 60^\circ = \frac{\sqrt{3}}{2}$)

Two spheres are projected at angles $30^\circ$ and $45^\circ$ with the horizontal. The maximum height reached by both is same. The ratio of their initial velocities is, $\left(\sin 45^\circ = \frac{1}{\sqrt{2}}, \sin 30^\circ = 0.5\right)$

Two balls A and B are projected at an angle of $45^\circ$ and $60^\circ$ respectively, so that the maximum heights reached are same for both. The ratio of initial velocity of projection of ball A to that for ball B is

$\left(\sin 30^\circ = \cos 60^\circ = \frac{1}{2},\ \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}},\ \sin 60^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2}\right)$

Two projectiles are fired with same initial speed from same point on ground at angles of $(45^\circ - \alpha)$ and $(45^\circ + \alpha)$, respectively, with the horizontal direction. The ratio of their maximum heights attained is:

Two projectiles are fired from ground with same initial speeds from same point at angles $(45^\circ + \alpha)$ and $(45^\circ - \alpha)$ with horizontal direction. The ratio of their times of flights is

Two balls with same mass and initial velocity, are projected at different angles in such a way that maximum height reached by first ball is 8 times higher than that of the second ball. $T_1$ and $T_2$ are the total flying times of first and second ball, respectively, then the ratio of $T_1$ and $T_2$ is

Projectiles A and B are thrown at angles of $45^\circ$ and $60^\circ$ with vertical respectively from top of a $400\text{ m}$ high tower. If their ranges and times of flight are same, the ratio of their speeds of projection $v_A : v_B$ is:

[Take $g = 10\text{ m s}^{-2}$]

Two bodies are projected from ground with same speeds $40\text{ m s}^{-1}$ at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of $60^\circ$, with horizontal then sum of the maximum heights, attained by the two projectiles, is ________ $\text{m}$. (Given $g = 10\text{ m s}^{-2}$)

Two projectiles are projected at $30^\circ$ and $60^\circ$ with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is:

The range of the projectile projected at an angle of 15$^\circ$ with horizontal is 50 m. If the projectile is projected with same velocity at an angle of 45$^\circ$ with horizontal, then its range will be

Two objects are projected with same velocity 'u' however at different angles $\alpha$ and $\beta$ with the horizontal. If $\alpha + \beta = 90^\circ$, the ratio of horizontal range of the first object to the 2nd object will be:

Two projectiles A and B are thrown with initial velocities of $40\text{ m/s}$ and $60\text{ m/s}$ at angles $30^\circ$ and $60^\circ$ with the horizontal respectively. The ratio of their ranges respectively is $\left(g = 10\text{ m/s}^{2}\right)$

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

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Two identical balls A and B thrown with same velocity 'u' at two different angles with horizontal attained the same range R. If A and B reached the maximum height h1 and h2 respectively, then $R = 4\sqrt{h_1 h_2}$

Reason R: Product of said heights.

$h_1 h_2 = \left(\frac{u^2 \sin^2\theta}{2g}\right) \cdot \left(\frac{u^2 \cos^2\theta}{2g}\right)$

Choose the correct answer:

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

Two projectiles P1 and P2 thrown with speed in the ratio $\sqrt{3}$: $\sqrt{2}$, attain the same height during their motion. If P2 is thrown at an angle of 60$^\circ$ with the horizontal, the angle of projection of P1 with horizontal will be:

A ball of mass m is thrown vertically upward. Another ball of mass $2m$ is thrown at an angle $\theta$ 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 $\frac{1}{x}$. The value of x is _____________.

Two projectiles are thrown towards each other at $15^\circ$ and $45^\circ$ angles respectively with the horizontal at the same speed. The difference between the horizontal distances traveled by the two projectiles is $80\text{ m}$. The initial speed of the projectiles is ______ $\text{m s}^{-1}$

[Given $g = 10\text{ m s}^{-2}$ ]

An object is projected in the air with initial velocity u at an angle $\theta$. 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 cases. The value of the angle of projection, at which the second object is projected, will be _________ degree.

The ranges and heights for two projectiles projected with the same initial velocity at angles 42$^\circ$ and 48$^\circ$ with the horizontal are R1, R2 and H1, H2 respectively. Choose the correct option:

Two particles are projected from the same point with the same speed u such that they have the same range R, but different maximum heights, h1 and h2. Which of the following is correct?

A shell is fired from a fixed artillery gun with an initial speed u such that it hits the target on the ground at a distance R from it. If t1 and t2 are the values of the time taken by it to hit the target in two possible ways, the product t1t2 is -

For angles of projection of a projectile at angle (45o $-\theta$) and (45o + $\theta$), the horizontal range described by the projectile are in the ratio of

A projectile can have the same range $R$ for two angles of projection. If $t_1$ and $t_2$ be the times of flights in the two cases, then the product of the two time of flights is proportional to

A projectile can have the same range 'R' for two angles of projection. If T1 and T2 be the time of flights in the two cases, then the product of the two time of flights is directly proportional to

Two projectiles of same mass and with same velocity are thrown at an angle 60o and 30o with the horizontal, then which will remain same

If a body $A$ of mass $M$ is thrown with velocity $v$ at an angle of $30^\circ$ to the horizontal and another body $B$ of the same mass is thrown with the same speed at an angle of $60^\circ$ to the horizontal, the ratio of horizontal range of $A$ to $B$ will be

If a body $A$ of mass $M$ is thrown with velocity $v$ at an angle of $30^\circ$ to the horizontal and another body $B$ of the same mass is thrown with the same speed at an angle of $60^\circ$ to the horizontal, the ratio of horizontal range of $A$ to $B$ will be