The acceleration of a certain particle is a function of time: a(t) = pt^2-qt^3, where p and q are constants. Initially, the velocity and position of the particle are zero.
(a) What is the velocity as a function of time?
(b) What is the position as a function of time?
v=∫(pt²-qt³)dt = p•t^3/3 -q•t^4/4,
s=∫v•dt=p•t^4/12 - q•t^5/20
To find the velocity as a function of time, we need to integrate the acceleration function with respect to time.
(a) Velocity as a function of time:
To find the velocity, we integrate the acceleration function a(t) = pt^2 - qt^3 with respect to time, t:
v(t) = ∫ a(t) dt = ∫ (pt^2 - qt^3) dt
Let's find the antiderivative of each term:
∫ (pt^2 - qt^3) dt = p ∫ t^2 dt - q ∫ t^3 dt
Using power rule integration, we have:
v(t) = p * (t^3 / 3) - q * (t^4 / 4) + C
Where C is the constant of integration.
(b) Position as a function of time:
To find the position, we integrate the velocity function v(t) with respect to time, t:
s(t) = ∫ v(t) dt = ∫ (p * (t^3 / 3) - q * (t^4 / 4) + C) dt
Let's find the antiderivative of each term:
s(t) = p * (∫ t^3 dt / 3) - q * (∫ t^4 dt / 4) + C * t + D
Using power rule integration:
s(t) = p * (t^4 / 12) - q * (t^5 / 20) + C * t + D
Where D is another constant of integration.
Therefore, the velocity as a function of time is v(t) = p * (t^3 / 3) - q * (t^4 / 4) + C, and the position as a function of time is s(t) = p * (t^4 / 12) - q * (t^5 / 20) + C * t + D.
To find the velocity as a function of time, we need to integrate the acceleration function with respect to time.
(a) Velocity as a function of time:
To find the velocity, we integrate the acceleration function. Let's start by integrating pt^2-qt^3 with respect to time.
∫ (pt^2 - qt^3) dt
The integral of pt^2 with respect to t is (p/3)t^3, and the integral of -qt^3 with respect to t is (-q/4)t^4.
Therefore, the velocity function v(t) is given by:
v(t) = ∫ (pt^2 - qt^3) dt
= (p/3)t^3 - (q/4)t^4 + C
Here, C is the constant of integration, representing the constant of velocity at time t = 0. Since the problem states that the initial velocity is zero, we can set C = 0. Hence, the velocity as a function of time is:
v(t) = (p/3)t^3 - (q/4)t^4
(b) Position as a function of time:
To find the position as a function of time, we integrate the velocity function obtained in part (a) with respect to time.
Let's integrate (p/3)t^3 - (q/4)t^4 with respect to t:
∫ ((p/3)t^3 - (q/4)t^4) dt
The integral of (p/3)t^3 with respect to t is (p/12)t^4, and the integral of -(q/4)t^4 with respect to t is -(q/20)t^5.
Therefore, the position function x(t) is given by:
x(t) = ∫ ((p/3)t^3 - (q/4)t^4) dt
= (p/12)t^4 - (q/20)t^5 + C
Here, C is the constant of integration, representing the constant of position at time t = 0. Since the problem states that the initial position is zero, we can set C = 0. Thus, the position as a function of time is:
x(t) = (p/12)t^4 - (q/20)t^5