$A$ body of mass $m$ moving along a straight line covers half the distance with a speed of $2 \,ms^{-1}$. The remaining half of the distance is covered in two equal time intervals with a speed of $3 \,ms^{-1}$ and $5 \,ms^{-1}$ respectively. The average speed of the particle for the entire journey is

$A$ particle moving along a straight line covers the first half of the distance with a speed of $3 \, m \, s^{-1}$. The other half of the distance is covered in two equal time intervals with speeds of $4.5 \, m \, s^{-1}$ and $7.5 \, m \, s^{-1}$ respectively. The average speed of the particle during the motion is:

$A$ train starts from rest from a station with acceleration $0.2 \, m/s^2$ on a straight track and then comes to rest after attaining maximum speed at another station due to retardation $0.4 \, m/s^2$. If the total time spent is half an hour,then the distance between the two stations is [Neglect the length of the train]. (in $, km$)

$A$ passenger arriving in a new town wishes to go from the station to a hotel located $10 \;km$ away on a straight road from the station. $A$ dishonest cabman takes him along a circuitous path $23 \;km$ long and reaches the hotel in $28 \;min$. What is
$(a)$ the average speed of the taxi,
$(b)$ the magnitude of average velocity? Are the two equal?

$A$ man in a car at location $Q$ on a straight highway is moving with speed $v$. He decides to reach a point $P$ in a field at a distance $d$ from the highway (point $M$) as shown in the figure. The speed of the car in the field is half of that on the highway. What should be the distance $RM$ so that the time taken to reach $P$ is minimum?

$A$ car travels from $A$ to $B$ (without changing direction). During the first part of the journey,its average speed is $V_1$,and for the second part of the journey,its average speed is $V_2$. The ratio of the path length of the second part to the path length of the first part is $\sqrt{\frac{V_2}{V_1}}$. In this case,the average speed of the total path is the ....... of the average speeds of both parts. Choose the correct option.