Relative Motion in One Dimension

Let us consider a boat that moves with a velocity $v_{b, w}=5 \mathrm{km}$${\mathrm{h}}^{-1}$ relative to water. At time $t=0$, the boat passes through a piece of cork floating in the water while moving downstream. If it turns back at time $t=t_1$, when and where does the boat meet the cork again? Assume $t_1=30 \min$.

A swimmer capable of swimming with velocity $v$ relative to water jumps in a flowing river having velocity $u$. The man swims a distance $d$ downstream and returns to the original position. Find out the time taken in complete motion.

Two cars $1$ and $2$ move with velocities $v_1$ and $v_2$, respectively, on a straight road in the same direction. When the cars are separated by a distance $d$, the driver of car $1$ applies brakes and the car moves with uniform retardation $a_1$. Simultaneously, car $2$ starts accelerating with $a_2$. If $v_1>v_2$, find the minimum initial separation between the cars to avoid collision between them.

An elevator is moving with an upward acceleration $a$. A coin is dropped from rest from the roof of the elevator, relative to you. After what time the coin will strike the base of the elevator?

A ball is thrown downwards with a speed of $20{ \mathrm{m} \mathrm{s}}^{-1}$ from the top of a building $150 \mathrm{m}$ high and simultaneously another ball is thrown vertically upwards with a speed of $30{ \mathrm{m} \mathrm{s}}^{-1}$ from the foot of the building. Find the time after which both the balls will meet. $\left(g=10{ \mathrm{m} \mathrm{s}}^{-2}\right)$

Two particles $A$ and$B$ are thrown vertically upward with velocity, $5 \mathrm{m}$ ${\mathrm{s}}^{-1}$ and $10 \mathrm{m}{ \mathrm{s}}^{-1}$, respectively $(g=10$$\mathrm{m} {\mathrm{s}}^{-2}$ ). Find separation between them after $1 \mathrm{s}$.

A car travelling at $60 \mathrm{km}{ \mathrm{h}}^{-1}$overtake another car travelling at $42 \mathrm{km}{ \mathrm{h}}^{-1}$. Assuming each car to be $5.0 \mathrm{m}$ long, find the time taken to overtake and the total road distance used for to overtake.

On a two-lane road, car $A$ is travelling at a speed of $36 \mathrm{km}{ \mathrm{h}}^{-1}$. Two cars $B$ and $C$ approach car $A$ in opposite directions with a speed of $54 \mathrm{km}{ \mathrm{h}}^{-1}$. At a certain instant, when the distance $AB$is equal to $AC$, both $1 \mathrm{km}, B$ decided to overtake $A$ before $C$ does. What minimum acceleration of car $B$ is required to avoid an accident?

A police van moving on a highway with a speed of $30 \mathrm{km}{ \mathrm{h}}^{-1}$ fires a bullet at a thief's car speeding away in the same direction with a speed of $192 \mathrm{km}{ \mathrm{h}}^{-1}$. If the muzzle speed of the bullet is $150 \mathrm{m} {\mathrm{s}}^{-1}$, with what speed does the bullet hit the thief's car?

Two towns $A$ and $B$ are connected by regular bus service with a bus leaving in either direction every $T$ $\min$. A man cycling with a speed of $20 \mathrm{km}{ \mathrm{h}}^{-1}$ in the direction $A$ to $B$ notices that a bus goes past him every $18$$\min$ in the direction of his motion and every $6 \min$ in the opposite direction. What is the period $T$ of the bus service and with what speed (assumed constant) do the buses ply on the road?

Suppose you are riding a bike with a speed of $10 \mathrm{m}{ \mathrm{s}}^{-1}$ due east relative to a person $A$ who is walking on the ground towards the east. If your friend $B$ walking on the ground due west measures your speed as $15 \mathrm{m}{ \mathrm{s}}^{-1}$, find the relative velocity between two reference frames $A$ and $B$.

A person walks up a stationary escalator in $t_1$ second. If he remains stationary on the escalator, then it can take him up in $t_2$ second. If the length of the escalator is $L$, then
(a) Determine the speed of man with respect to the escalator.
(b) Determine the speed of the escalator.
(c) How much time would it take him to walk up the moving escalator?

A bird flies to and fro between two cars which move with velocities $v_1$ and $v_2$. If the speed of the bird is $v_3$ and the initial distance of separation between the cars is $d$, find the total distance covered by the bird till the cars meet.

A car $A$ moves with velocity $15 \mathrm{m}{ \mathrm{s}}^{-1}$ and $B$ with velocity $20 \mathrm{m} {\mathrm{s}}^{-1}$ are moving in opposite directions as shown in the figure. Find the relative velocity of $B$ with respect to $A$ and $A$ with respect to $B$.

A car $A$ moves with velocity $20{ \mathrm{m} \mathrm{s}}^{-1}$ and the car $B$ with velocity $15{ \mathrm{m} \mathrm{s}}^{-1}$ as shown in the figure. Find the relative velocity of $B$ with respect to $A$ and $A$ with respect to $B$.

Two parallel rail tracks run north to south. The train $A$ moves north with a speed of $54 \mathrm{km} {\mathrm{h}}^{-1}$ and the train $B$ moves south with a speed of $90 \mathrm{km} {\mathrm{h}}^{-1}$. A monkey running on the roof of the train $A$ against its motion (with its velocity of $18 \mathrm{km} {\mathrm{h}}^{-1}$ with respect to the train $A$). What is the velocity of the monkey, $\left(\mathrm{a}\right)$ as observed by an observer on the ground? $\left(\mathrm{b}\right)$ as observed by an observer moving in train $B$?

A railroad flatcar is travelling to the right at a speed of $15 \mathrm{m} {\mathrm{s}}^{-1}$ relative to an observer sitting on the ground. Someone is riding a scooter on the flatcar (as shown in the figure). The velocity of a scooter with respect to a car is $10 \mathrm{m} {\mathrm{s}}^{-1}$. What is the velocity of the scooter relative to the observer on the ground in case $\left(\mathrm{a}\right)$ and case $\left(\mathrm{b}\right)$ as shown in the figure?

Two stones are thrown up simultaneously from the edge of a cliff $240 \mathrm{m}$ high with initial speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$ and $40 \mathrm{m} {\mathrm{s}}^{-1}$, respectively. Which of the following graphs best represent the time variation of relative position of the second stone with respect to the first? (Assume that the stones do not rebound after hitting the ground and neglect air resistance. Take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$. The figures are schematic and not drawn to scale.)

A man swimming downstream overcomes a float at a point $M$. After travelling distance $D$, he turned back and passed the float at a distance of $\frac{D}{2}$ from the point $M$. Then the ratio of the speed of the swimmer with respect to still water to the speed of the river will be

A police jeep is chasing a culprit going on a motorbike. The motorbike crosses a turning at a speed of $72 \mathrm{km} {\mathrm{h}}^{-1}$. The jeep follows it at a speed of $90 \mathrm{km} {\mathrm{h}}^{-1}$, crossing the turning $10 \mathrm{s}$ later than the bike. Assuming that they travel at constant speeds, how far from the turning will the jeep catch up with the bike? (In $\mathrm{km}$)