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
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.