A particle is moving so that its velocity, v(t) = t-64/t^2 for 2≤t≤6. find the total distance traveled by the particle.
distance= integral v(t) dt for 2 to 6
= INT (t-64/t^2)dt= t^2+64/t
put in the limits 6,2
do you mean
(t-64)/t^2 maybe
or
t - (64/t^2) as you typed it
???
its --> t - (64/t^2)
could you please solve it..
Ok, that is what I did.
To find the total distance traveled by the particle, we need to integrate the absolute value of the velocity function over the given time interval.
First, let's find the interval over which the particle is moving in the positive direction:
To determine this, we need to find when the velocity is positive (moving in the positive direction). So, we solve the equation v(t) > 0:
t - 64/t^2 > 0
Multiplying through by t^2 to clear the denominator, we get:
t^3 - 64 > 0
Taking the cube root of both sides to isolate t, we have:
t > ∛64
Simplifying, we find t > 4.
So, the particle is moving in the positive direction for values of t greater than 4.
Next, let's find the interval over which the particle is moving in the negative direction:
To determine this, we need to find when the velocity is negative (moving in the negative direction). So, we solve the equation v(t) < 0:
t - 64/t^2 < 0
Multiplying through by t^2 to clear the denominator, we get:
t^3 - 64 < 0
Taking the cube root of both sides to isolate t, we have:
t < ∛64
Simplifying, we find t < 4.
So, the particle is moving in the negative direction for values of t less than 4.
Now that we have determined the intervals in which the particle is moving in the positive and negative directions, we can calculate the total distance traveled.
The total distance traveled is given by the sum of the distances traveled in each direction. Since the particle is moving in the positive direction for t > 4 and in the negative direction for t < 4, we need to evaluate the integral of the absolute value of the velocity function over each of these intervals separately.
For the positive direction (t > 4), we integrate the absolute value of the velocity function v(t) from 4 to 6:
∫[4 to 6] |t - 64/t^2| dt
For the negative direction (t < 4), we integrate the absolute value of the velocity function v(t) from 2 to 4:
∫[2 to 4] |t - 64/t^2| dt
Evaluating these integrals will give us the total distances traveled in each direction. Adding these two distances together will give us the total distance traveled by the particle.