Time, Speed and Distance

A train of length L move with a constant speed $V_t$. A person at the back of the train fires a bullet at time $t=0$ towards a target which is at a distance of D(at time $t=0$) from the front of the train(on the same direction of motion). Another person at the front of the train fires another bullet at time $t=T$ towards the same target. Both bullets reach the target at the same time. Assuming the speed of the bullets $V_b$ are same, the length of the train is?

A car is moving with at a constant speed of 60 km ${ h }^{ -1 }$ on a straight road. Looking at the rear view mirror, the driver finds that the car following him is at a distance of 100 m and is approaching with a speed of 5 km ${ h }^{ -1 }$. In order to keep track of the car in the rear, the driver begins to glance alternatively at the rear and side mirror of his car after every 2 s till the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correct?

A train leaves station 'A' to station 'B'. The train travels straight without any halts between the stations. During the first and last $200$ m of its journey, the train has uniform acceleration and retardation both equal to $\displaystyle 1\quad { ms }^{ -1 }$ respectively. For the rest of the journey, the train maintains uniform speed Calculate the average speed of the train, given the distance between two stations is 4 km.

Two swimmers start a race. One who reaches the point $C$ first on the other bank wins the race. $A$ makes his strokes in a direction of $37^{o}$ to the river flow with velocity $5\ km/hr$ relative to water. $B$ make his strokes in a direction $127^{o}$ to the river flow with same relative velocity. River is flowing with speed of $2\ km/hr$ and is $100\ m$ wide. speeds of $A$ and $B$ on the ground are $8 \ km/hr$ and $6 \ km/hr$ respectively .

The correct match is: List - I List - II a) A body covers first half of distance with a speed $V_{1}$ and second half of distance with a speed $V_{2}$ e) Average velocity is $\sqrt\frac{gh}{2}$ b) A body covers first half of the time with a speed $V_{1}$ and second half of the time with a speed $V_{2}$ f) Average speed is $\frac{v_{1}+v_{2}}{2}$ c) A body is projected vertically up from ground with certain velocity.Considering its total motion, g) Average speed is $\frac{2V_{1}V_{2}}{V_{1}+V_{2}}$ d) A body freely is released from a height h h) Average velocity is zero

A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is $0.6$ km north of the intersection and the car is $0.8$ km to the east, the police determine with radar that the distance of the car is increasing at $20$ km/h. Suppose that the cruiser is moving at $60$ km/h at the instant of measurement. $\displaystyle \frac{d^{2}s}{dt^{2}}$ equal to at the moment in question if $\displaystyle \frac{d^{2}x}{dt^{2}}=-40$ and $\displaystyle \frac{d^{2}y}{dt^{2}}=50$(in km/h$^{2}$)

Lovish has $6$ hours to spend in Ha Ha Tonka State Park. He plans to drive around the park at an average speed of $20$ miles per hour, looking for a good trail to hike. Once he finds a trial he likes, he will spend the remainder of his time hiking it. He hopes to travel more than $60$ miles total while in the park. If he hikes at an average speed of $1.5$ miles per hour, which of the following systems of inequalities can be solved for the number of hours Lovish sends driving d, and the number of hours he spends hiking, h while he is at the park?