If the velocity function of an object is v(t) = -5t4 - 20t + 100, find the displacement function given that the displacement is 50 ft when time is 1 second
v(t) = -5t^4 - 20t + 100
s(t) = -t^5 - 10t^2 + 100t + c
you want c, so just plug in s(1)=50 and crank it out.
To find the displacement function, we need to integrate the velocity function with respect to time. The displacement function, denoted as s(t), represents the position of the object relative to a reference point.
Given that the displacement is 50 ft when time is 1 second, we have the initial condition s(1) = 50.
To find the displacement function, we need to integrate the velocity function v(t) = -5t^4 - 20t + 100 with respect to time:
∫ v(t) dt = ∫ (-5t^4 - 20t + 100) dt
By integrating term by term, we have:
∫ -5t^4 dt - ∫ 20t dt + ∫ 100 dt
Taking the integral of each term:
-∫ 5t^4 dt - ∫ 20t dt + ∫ 100 dt
To integrate each term:
- (5/5) * ∫ t^4 dt - 20 * ∫ t dt + 100 * ∫ dt
Simplifying further:
- t^5 + 10t^2 + 100t + C
Where C is the constant of integration.
Now we can substitute t = 1 into the displacement function to find the constant of integration:
s(1) = - (1)^5 + 10(1)^2 + 100(1) + C = 50
Simplifying this equation:
-1 +10 +100 + C = 50
109 + C = 50
C = 50 - 109
C = -59
Therefore, the displacement function is:
s(t) = -t^5 + 10t^2 + 100t - 59
To find the displacement function, we need to integrate the velocity function with respect to time. The displacement function is the antiderivative of the velocity function.
Given that the velocity function is v(t) = -5t^4 - 20t + 100, we integrate it to find the displacement function.
∫v(t) dt = ∫(-5t^4 - 20t + 100) dt
Let's integrate each term separately:
∫(-5t^4) dt = - (5/5)t^5 + C1 = -t^5 + C1
∫(-20t) dt = -20(1/2)t^2 + C2 = -10t^2 + C2
∫(100) dt = 100t + C3
Adding up the three obtained integrals, we get:
∫v(t) dt = -t^5 - 10t^2 + 100t + C
This is the displacement function, where C represents the constant of integration. The constant of integration accounts for any unknown initial conditions or reference points.
To find the displacement when time is 1 second, we substitute t = 1 into the displacement function and set it equal to 50, as given:
-t^5 - 10t^2 + 100t + C = 50
Simplifying this equation would yield the value of the constant of integration, C.