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
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"