A particle is moving along a line so that if v feet per second is the velocity of the particle at t second, then v= t^2-t+1/((t+2)^2(t^2+1)). Find a formula for the distance traveled by the particle from the time when ๐ก = 0 ๐ก๐ ๐กโ๐ ๐ก๐๐๐ ๐คโ๐๐ ๐ก = t sub 1
that would, of course, be
โซ[0,t_1] v(t) dt
You'll have to use partial fractions to do this one.
To find the formula for the distance traveled by the particle from ๐ก = 0 to ๐ก = ๐กโ, we need to integrate the velocity function with respect to time over that interval.
Given that the velocity function v(t) = tยฒ - t + 1 / ((t + 2)ยฒ(tยฒ + 1)), we'll find the antiderivative of v(t) and then evaluate it at ๐ก = ๐กโ and ๐ก = 0.
Step 1: Find the antiderivative of v(t)
To find the antiderivative, we integrate each term separately:
โซ(tยฒ - t + 1) / ((t + 2)ยฒ(tยฒ + 1)) dt
Let's break it down:
โซ(tยฒ - t + 1) / ((t + 2)ยฒ(tยฒ + 1)) dt
= โซtยฒ / ((t + 2)ยฒ(tยฒ + 1)) dt - โซt / ((t + 2)ยฒ(tยฒ + 1)) dt + โซ1 / ((t + 2)ยฒ(tยฒ + 1)) dt
Step 2: Evaluate the antiderivative at ๐ก = ๐กโ and ๐ก = 0
Let F(t) be the antiderivative of v(t):
F(t) = โซtยฒ / ((t + 2)ยฒ(tยฒ + 1)) dt - โซt / ((t + 2)ยฒ(tยฒ + 1)) dt + โซ1 / ((t + 2)ยฒ(tยฒ + 1)) dt
Now, evaluate F(t) at ๐ก = ๐กโ and ๐ก = 0:
Distance traveled = F(๐กโ) - F(0)
To find the specific formula for the distance traveled, we'll need to calculate each of the integrals separately, evaluate them at ๐ก = ๐กโ and ๐ก = 0, and then subtract F(0) from F(๐กโ).
Unfortunately, finding an explicit formula for the distance traveled may not be feasible without specific values for ๐กโ. However, you can calculate the distance traveled by evaluating the definite integral numerically using numerical methods or software such as a graphing calculator or computer algebra system (CAS).