A particel is moving with the given data. Find the position of the particle.
v(t)= sin t - cos t , s(0)= 0
position is the integral
p(t)=Defintegral (v(t) dt)
if the limits are 0 to t
then p(t)=-cost-sint -(-cos0-sin0)) check all that, I did it in my head.
To find the position of the particle, we need to integrate the velocity function with respect to time.
Given:
v(t) = sin t - cos t
s(0) = 0
Integrating v(t) will give us the position function s(t).
∫v(t) dt = ∫(sin t - cos t) dt
To integrate the function sin t, we use the fact that the integral of sin t is -cos t. Similarly, the integral of cos t is sin t.
∫(sin t - cos t) dt = -cos t - sin t + C
where C is the constant of integration.
Now we need to determine the value of C using the initial condition, s(0) = 0.
Plugging in t = 0 into the position function, we get:
- cos 0 - sin 0 + C = 0
- (1) - (0) + C = 0
- 1 + C = 0
C = 1
Therefore, the position function s(t) = -cos t - sin t + 1.