Applications of Relative Projectile

Q. A particle moving in a circle of radius $R$ with a uniform speed takes a time $T$ to complete one revolution. If this particle were projected with the same speed at an angle $\theta$ to the horizontal, the maximum height attained by it is $4R$. The angle of projection, $\theta$ is then given by :

1. $\theta ={\sin }^{-1}{\left(\frac{2g{T}^2}{{\pi }^{2}R}\right)}^{1/2}$
2. $\theta ={\cos }^{-1}{\left(\frac{g{T}^2}{{\pi }^{2}R}\right)}^{1/2}$
3. $\theta ={\cos }^{-1}{\left(\frac{{\pi }^{2}R}{g{T}^2}\right)}^{1/2}$
4. $\theta ={\sin }^{-1}{\left(\frac{{\pi }^{2}R}{g{T}^2}\right)}^{1/2}$

Q. The maximum speed with which a vehicle can negotiate a curved road, which is banked at the angle, $\theta ={\tan }^{-1}\left(0.24\right)$ is $54\text{km/hr}$. If another road is flat and the vehicle has to negotiate a curve with the same maximum speed, coefficient of friction between road and tyre should be
(Radius of both the road is same)

1. $0.35$
2. $0.24$
3. $0.8$
4. $0.5$

Q. An object is projected with speed $50\text{m/s}$ at an angle ${53}^{\circ }$ with horizontal from ground. The radius of curvature of its trajectory at $t=1\text{sec}$ after projection will be:
$($Take $g=10{\text{m/s}}^2)$

1. $90\sqrt{2}\text{m}$
2. $180\sqrt{2}\text{m}$
3. $90\text{m}$
4. $125\text{m}$

Q. A ball is projected from the ground at a speed of $10m{s}^{-1}$ making an angle ${30}^{\circ }$ with the horizontal. Another ball is simultaneously released from a point on the vertical line along the maximum height of the projectile. Both the balls collide at the maximum height of the projectile. What was the initial height of the freely falling body?( Assume $g=10m{s}^{-2}$)

1. $5m$
2. $10m$
3. $4m$
4. $2.5m$

Q. In the figure shown, the two projectiles are fired simultaneously. Find the minimum distance between them during their flight?

Q. A particle (A) moves due north at 3 km/h, another particle (B) due west at 4 km/h. The relative velocity of A with respect to B is $(tan{37}^{\circ }=3/4)$

1. $5km/h,{37}^{\circ }$north of east
2. $5km/h,{37}^{\circ }$ east of north
3. $5\sqrt{2}km/h,{53}^{\circ }$ east of north
4. $5\sqrt{2}km/h,{53}^{\circ }$ north of east

Q. A particle is projected from the ground at an angle of ${60}^{\circ }$ with the horizontal with speed $u=20\text{m/s}$. The radius of curvature of the path of the particle, when its velocity makes an angle ${30}^{\circ }$ with the horizontal is
$(g=10{\text{m/s}}^2)$

1. $10.6\text{m}$
2. $12.8\text{m}$
3. $15.4\text{m}$
4. $24.2\text{m}$

Q. A large rectangular box falls vertically with an acceleration a. A toy gun fixed at A and aimed towards C fires a particle P. Then

1. P will hit C if a = g
2. P will hit the roof BC if a > g
3. P will hit the wall CD if a < g
4. May be either (a), (b) or (c), depending on the speed of projection of P

Q. A student is standing on a train travelling along a straight horizontal track at a speed of $10\text{m/s}$. The student throws a ball into the air along a path, that he sees to make an initial angle of ${60}^{\circ }$ with the horizontal along the track. The professor standing on the ground observes the ball to rise vertically, the maximum height reached by the ball is

Q. Two particles are projected simultaneously from the towers of same height as shown in the figure with velocities $V_A=20\text{m/s}$ and $V_B=10\text{m/s}$ respectively. They collide in air after 0.5 seconds. Find ${}^{\prime }{\theta }^{\prime }$. Answer in degrees

Q. A boy is standing inside a train moving with a constant velocity of magnitude $10m/s$. He throws a ball vertically up with a speed of $5m/s$ relative to the train. Find the radius of curvature of the path of the ball just at the time of projection. (Take $g=10m/s^2$)

1. $14m$
2. $10m$
3. $7m$
4. $5m$

Q.

A car is travelling on a highway at a speed of 25 m/s along the x-axis. A passenger in a car throws a ball at an angle ${37}^{\circ }$ with horizontal in a plane perpendicular the motion of the car. The ball is projected with a speed of 10 m/s relative to the car. What may be the initial velocity of the ball in unit vector notation?

1. 

2. 

3. 

4. 

Q. An aeroplane is travelling horizontally at a height of $2000\text{m}$ from the ground. The aeroplane, when at a point $\text{P}$, drops a bomb to hit a stationary target $Q$ on the ground. In order that the bomb hits the target, what angle $\theta$ must the line $\text{PQ}$ make with the vertical ?
$[g=10{\text{m/s}}^2]$

1. ${30}^{\circ }$
2. ${37}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. A man is walking on a road with a velocity of 3 Km/h when suddenly, it starts raining velocity of rain is 10 Km/h in vertically downward direction, relative velocity of the rain with respect to man is:

1. $\sqrt{13}$ Km/hr
2. $\sqrt{7}$ Km/hr
3. $\sqrt{109}$ Km/hr
4. 13 Km/h

Q. Rain is falling vertically downwards with a velocity $3km/hr$. A man walks in the rain with a velocity of $4km/hr$. The rain drop will fall on the man with a velocity of

1. $5km/hr$
2. $4km/hr$
3. $1km/hr$
4. $3km/hr$

Q. A ball is projected upwards from the top of a tower with a velocity $50\text{m/s}$ making an elevation angle ${30}^{\circ }$ with the horizontal. The height of the tower is $70\text{m}$. Find time taken by the ball to reach the ground from the instant of throwing.

1. $2\text{sec}$
2. $5\text{sec}$
3. $7\text{sec}$
4. $9\text{sec}$

Q. Two stones are projected simultaniously from a tower at different angles of projection with same speed $u$. The distance between two stones is increasing at constant rate $u$. Then the angle between the initial velocity vectors of the two stone is:

1. ${90}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. Two particles $A$ and $B$ are projected in air. $A$ is thrown with a speed of $60m/sec$ and $B$ with a speed of $80m/sec$ as shown in the figure.
What is the separation between them after $2sec$?

1. $100m$
2. $150m$
3. $200m$
4. $250m$

Q. A train has to negotiate a curve of radius $2000\text{m}$. By how much should be the outer rail be raised with respect to the inner rail for a safe travelling speed of $72{\text{kmh}}^{-1}$?
The distance between the rails is $1\text{m}$.
$($Take $g=10{\text{m/s}}^2)$

1. $1\text{cm}$
2. $3\text{cm}$
3. $2\text{cm}$
4. $4.5\text{m}$

Q. A simple pendulum of length $l$ and mass (bob) $m$ is suspended vertically. The string makes an angle $\theta$ with the vertical. The restoring force acting on the pendulum is:

1. $mg\tan \theta$
2. $-mg\sin \theta$
3. $mg\sin \theta$
4. $-mg\cos \theta$

Q. An object is thrown along a direction inclined at an angle of ${45}^o$ with the horizontal direction. The horizontal range of the particle is equal to

1. Twice the vertical height
2. Vertical height
3. Thrice the vertical height
4. Four times the vertical height

Q. Suppose two particles, 1 and 2, are projected in vertical plane simultaneously.

Their angles of projection are ${30}^{\circ }$ and $\theta$ respectively with the horizontal. If they collide
after a time $t$ in air, then

1. $\theta ={\sin }^{-1}\left(\frac{4}{5}\right)$ and they will have same speed just before the collision.
2. $\theta ={\sin }^{-1}\left(\frac{4}{5}\right)$ and they will have different speed just before the collision.
3. $x<1280\sqrt{3}-960\text{m}$
4. it is possible that the particle collide when both of them are at their highest point.

Q. A car of mass $m$ is moving on a horizontal circular track of radius $r$. The time taken to finish one complete revolution is $\frac{\pi }{2}\text{s}$. If the velocity of the car at an instant $t$ is given by $v=5\sin 2t$, then the magnitude of tangential acceleration of the car at the time when it finishes a quarter circle is $p$. The value of $p^2$ is (No slipping between the car and track)

Q. A particle is thrown vertically upward. Its velocity at half of the height is 10 m/s. Then the maximum height attained by it (g = 10 ${m/s}^2$) :

1. 8 m
2. 20 m
3. 10 m
4. 16 m

Q. A ball is projected with a velocity $20\sqrt{3}m/s$ at angle ${60}^o$ to the horizontal. The time interval after which the velocity vector will make an angle ${30}^o$ to the horizontal is (in seconds)

1. $3$
   
2. $2$
   
3. $5$
   
4. $9$
   

Q. A body is projected vertically upward with speed $40$ m/s. The distance travelled by body in the last second of upward journey is [ take $g=9.8m/s^2$ and neglect effect of air resistance]

1. $4.9m$
2. $9.8m$
3. $12.4m$
4. $19.6m$

Q. A projectile is projected at an angle $60$ degree with horizontal with speed $10m/s$. The minimum radius of curvature of the trajectory described by the projectile is (in $m$) :

1. $2.55$
2. $3$
3. $6$
4. $9$

Q. A large rectangular box falls vertically with an acceleration a. A toy gun fixed at A and aimed towards C fires a particle P. Then

1. P will hit C if a = g
2. P will hit the roof BC if a > g
3. P will hit the wall CD if a < g
4. May be either (a), (b) or (c), depending on the speed of projection of P

Q. A body is projected at $t=0$ with a velocity $10{\text{ms}}^{-1}$ at an angle ${60}^{\circ }$ with the horizontal. The radius of curvature of its trajectory at $t=1$ is $R$. Neglecting air resisteance and taking acceleration due to gravity $g=10{\text{m/s}}^2$, the value of $R$ is:

1. $10.3\text{m}$
2. $2.8\text{m}$
3. $2.5\text{m}$
4. $5.1\text{m}$

Q. A very broad elevator is going up vertically with a constant acceleration of $2m/s^2$. At the instant when its velocity is $4m/s$ a ball is projected from the floor of the lift with a speed of $4m/s$, relative to the floor at an elevation of ${30}^o$. The time taken by the ball to return the floor is
($g=10m/s^2$)

1. $\frac{1}{2}s$
2. $\frac{1}{3}s$
3. $\frac{1}{4}s$
4. $1s$