A particle starts from rest and has a constant acceleration of #4m/s^2# for #4sec#. It then retards uniformly for next #8sec# and comes to rest. Average speed of the particle during the motion is?
A: #16m/s#
B: #8m/s#
C: #24m/s#
D: None of the above
A:
B:
C:
D: None of the above
1 Answer
(B)
Explanation:
We're asked to find the average speed of a particle during a motion.
The equation for average speed is
#v_"av" = "distance traveled"/(Deltat)#
where
We know it uniformly accelerated from rest at a rate of
#a_x = 4# #"m/s"^2#
for
#t = 4# #"s"#
We can use the kinematics equation
#ul(Deltax = v_(0x)t + 1/2a_xt^2#
to find the distance it travels
Here, since it started from rest, the initial velocity
#ul(Deltax = 1/2a_xt^2#
Plugging in known values:
#color(red)(Deltax_1) = 1/2(4color(white)(l)"m/s"^2)(4color(white)(l)"s")^2 = color(red)(ul(32color(white)(l)"m"#
For the second part of this motion, we're given that the acceleration is constant (and unknown) and that it comes to rest in
We need to find the velocity of the particle after it has finished its first acceleration, using the equation
#Deltax_1 = ((v_(0x) + v_x)/2)t#
where
#color(red)(32color(white)(l)"m") = ((0 + v_x)/2)(4color(white)(l)"s")#
#v_x = color(green)(ul(16color(white)(l)"m/s"#
(you could've also used the equation
This value represents the initial velocity of the particle as it begins to negatively accelerate. We can now use the same equation
#Deltax_2 = ((v_(0x) + v_x)/2)t#
to find the distance traveled
Here,

#v_x# , the final velocity, is#0# (it comes to rest) 
#v_(0x)# is#color(green)(16color(white)(l)"m/s")# 
#t# is#8# #"s"# :
#color(purple)(Deltax_2) = ((color(green)(16color(white)(l)"m/s") + 0)/2)(8color(white)(l)"s") = color(purple)(ul(64color(white)(l)"m"#
The total distance traveled is
#"distance traveled" = color(red)(Deltax_1) + color(purple)(Deltax_2)#
#= color(red)(32color(white)(l)"m") + color(purple)(64color(white)(l)"m") = color(orange)(ul(96color(white)(l)"m"#
And the time
#Deltat = 4# #"s"# #+ 8# #"s"# #= ul(12color(white)(l)"s"#
Thus, the average speed of the particle is
#v_"av" = (color(orange)(96color(white)(l)"m"))/(12color(white)(l)"s") = color(blue)(ulbar(stackrel(" ")(" "8color(white)(l)"m/s"" "))#
Therefore, option