Acceleration

On a frictionless horizontal surface, assumed to be the x-y plane, a small trolley A is moving along a straight line parallel to the y-axis (see figure) with a constant velocity of $\left(\sqrt{3}-1\right)m{s}^{-1}$. At a particular instant, when the line OA makes an angle of $45°$ with the x-axis, a ball is thrown along the surface from the origin O. Its velocity makes an angle $\varphi$ with the x-axis and it hits the trolley. The motion of the ball is observed from the frame of trolley. Calculate the angle $\theta$ made by the velocity vector of the ball with the x-axis in this frame.

A ball is thrown from a point with a speed $v_0$  at an angle of projection $\theta$ . from the same point and at the same instant, a person starts running with a constant speed  $\frac{v_0}{2}$ to catch the ball. will the person be able to catch the ball? If yes, what should be the angle of projection?

Four persons $K$, $L$, $M$, $N$ are initially at the four corners of a square of side $d$. Each person now moves with a uniform speed  $v$ in such a way that $K$ always moves directly towards $L$, L directly towards $M$, $M$ directly towards $N$, and $N$ directly towards $K$. The four persons will meet at a time

A swimmer crosses a river with minimum possible time $10$ second. And when he reaches the other end he starts swimming in the direction towards the point from where he started swimming. Keeping the direction fixed the swimmer crosses the river in $15 \sec$. The ratio of speed of swimmer with respect to water and the speed of river flow is (Assume constant speed of river & swimmer):

A person of height $1.6 \mathrm{m}$ is walking away from a lamp post of height $4 \mathrm{m}$ along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60 \mathrm{cm} {\mathrm{s}}^{-1}$, the speed of the tip of the person’s shadow on the ground with respect to the person is _____ $\mathrm{cm} {\mathrm{s}}^{-1}$.

A particle is moving in a straight line such that its velocity is increasing at $5 {\mathrm{ms}}^{-1}$ per meter. The acceleration of the particle is _____ ${\mathrm{ms}}^{-2}$ at a point where its velocity is $20 {\mathrm{ms}}^{-1}$.

An airplane moves in a northwesterly direction at $125 \mathrm{mi}/\mathrm{hr}$ relative to the ground, due to the fact there is a westerly wind of $50 \mathrm{mi}/\mathrm{hr}$ relative to the ground. How fast and in what direction would the plane have traveled if there were no wind?

A passenger in an open car travelling at $30 \mathrm{m} {\mathrm{s}}^{-1}$ throws a ball out over the bonnet. Relative to the car the initial velocity of the ball is $20 \mathrm{m} {\mathrm{s}}^{-1}$ at $60°$ to the horizontal. The angle of projection of the ball with respect to the horizontal road will be:

A gun mounted on the top of a moving truck is aimed in the backward direction at angle of $30°$ to the vertical. If the muzzle velocity of the gun is $4 \mathrm{m} {\mathrm{s}}^{-1}$, the speed of the truck to send the bullet vertically up is

An aeroplane travels at the speed of $400 \mathrm{km} {\mathrm{h}}^{-1}$. It ejects its products of combustion at the speed of $1200 \mathrm{km} {\mathrm{h}}^{-1}$ with respect to an observer on the ground. The speed of ejected product of combustion relative to the aeroplane is

The wind is blowing from the south at $10 \mathrm{m}/\mathrm{s}$but to a cyclist, it appears blowing from the east at $10 \mathrm{m}/\mathrm{s}$. The velocity of the cyclist is 

A particle moves in a straight line so that distance $s=\sqrt{t}$ where $t$ is time, then the acceleration is proportional to

A bird is flying in a direction $30°$ east of north with a speed of $5\mathrm{kmph}$ and a cyclist going on the road towards east at a speed of $10\mathrm{kmph}$. The velocity of the bird observed by the cyclist is

A shell is fired vertically from a cannon which is on a vehicle travelling horizontally at constant speed $u=36 \mathrm{km} {\mathrm{h}}^{-1}$. The shell leaves the cannon with a velocity $10 \mathrm{m} {\mathrm{s}}^{-1}$ relative to the cannon. The shell will land

Car A travelling towards the positive x-axis with speed $10 \mathrm{m}/\mathrm{s}$ where as car B is travelling towards the positive y-axis with speed $10 \mathrm{m}/\mathrm{s}$. Find the angle made by the velocity of car A as seen by car B with positive -axis

A particle $A$ is moving towards North and acceleration of $5 \mathrm{m} {\mathrm{s}}^{-2}$ and particle $B$ is moving North-East direction with an acceleration of $5\sqrt{2}\mathrm{m} {\mathrm{s}}^{-2}$. Find relative acceleration of particle $A$ with respect to particle $B$.

A train is accelerating towards north at the rate of $4.9 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.$. Find the acceleration of a bolt which falls from the ceiling relative to  the Earth?

A train is accelerating towards north at the rate of $4.9 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.$ . Find the acceleration of a bolt which falls from the ceiling relative to the train.

A cat is climbing a vertical pole with a speed of $4 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ and a dog is running towards the pole with a speed of $8 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$. What is the velocity of the dog relative to the cat ?

The wind is blowing from the south at $10 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$, but to a cyclist it appears to be blowing from the east at $10 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$. Determine the velocity of the cyclist in vector form.