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

1 Answer
Nathan L.
Aug 4, 2017

(B) #8# #"m/s"#

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 #Deltat# is the time interval

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 #Deltax# during this acceleration.

Here, since it started from rest, the initial velocity #v_(0x)# is #0#, leaving us with

#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 #8# #"s"#.

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 #v_x# is the velocity in question, and #v_(0x)# is still #0# as before:

#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 #v_x = v_(0x) + a_xt# to find the velocity.)

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 #Deltax#.

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# is

#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 #color(blue)("B"# is correct.

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