25/3 m/s is ans? a particle is moving along positive X axis with uniform acceleration -2m/s. its initial velocity is 20m/s. calculate out average speed after 15 second

1 Answer
Nathan L.
Jun 8, 2017

#25/3"m"/"s"#

Explanation:

We're asked to calculate the average speed of an object moving with constant acceleration along the #x#-axis.

The formula for average speed is

#overbrace(v_(av-x))^"speed" = "total distance traveled"/(Deltat)#

We'll take its starting point to be #x = 0# to simplify things.

We need to find its location after #15# #"s"# with the given acceleration and initial velocity. We can use the equation

#x = x_0 + v_(0x)t + 1/2a_xt^2#

to find the object's new position, and thus its displacement, after #15# #"s"#. Plugging in known variables, we have

#x = (20"m"/"s")(15"s") + 1/2(-2"m"/("s"^2))(15"s")^2 = color(red)(75# #color(red)("m"#

Since the acceleration is negative, we don't know from this if this is the displacement after it started to turn around or before. To check, we can find its total displacement when it starts to turn around (i.e. when its velocity is #0#) using the equation

#(v_x)^2 = (v_(0x))^2 + 2a_x(Deltax)#

Plugging in #0# for #v_x#, we have

#0 = (20"m"/"s")^2 + 2(-2"m"/("s"^2))(Deltax)#

#(4"m"/("s"^2))(Deltax) = 400("m"^2)/("s"^2)#

#Deltax = 100# #"m"#

Now we know how far it goes before it turns around. To check whether the displacement in #15# #"s"# was before or after this, we need to find the time #t# when it starts to turn around. If the calculated time is before #15# #"s"#, then the object was going backward at #t = 15# #"s"#. If the time is after #15# #"s"#, then it was still going forward.

To find the time #t# when the velocity is #0#, we can use the equation

#v_x = v_(0x) + a_xt#

Plugging in #0# for #v_x#, we have

#0 = 20"m"/"s" - (2"m"/("s"^2))t#

#t = 10# #"s"#

Therefore, the object was on its way backward at #t = 15# #"s"#.

To find the total distance traveled, we take the #100# #"m"# and add it to the distance it traveled backward, which is #100# #"m" - 75# #"m" = 25# #"m"#.

The total distance traveled is thus

#100# #"m" + 25# #"m" = color(blue)(125# #color(blue)("m"#

Finally, going back to the average speed formula, the average speed of the object over the time interval is

#overbrace(v_(av-x))^"speed" = (125"m")/(15"s")#

which simplifies to

#color(green)(25/3"m"/"s"#, or #color(green)(8.33"m"/"s"#

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