A body, having initial velocity u metre per second ,is accelerating uniformly along a straight path. After certain time interval it's velocity becomes v metre per second. What is the average velocity of the body for that time interval?
Answers
Let’s begin with a particle with an acceleration a(t) is a known function of time. Since the time derivative of the velocity function is acceleration,
ddtv(t)=a(t),ddtv(t)=a(t),
we can take the indefinite integral of both sides, finding
∫ddtv(t)dt=∫a(t)dt+C1,∫ddtv(t)dt=∫a(t)dt+C1,
where C1 is a constant of integration. Since ∫ddtv(t)dt=v(t)∫ddtv(t)dt=v(t), the velocity is given by
v(t)=∫a(t)dt+C1.v(t)=∫a(t)dt+C1.
Similarly, the time derivative of the position function is the velocity function,
ddtx(t)=v(t).ddtx(t)=v(t).
Thus, we can use the same mathematical manipulations we just used and find
x(t)=∫v(t)dt+C2,x(t)=∫v(t)dt+C2,
where C2 is a second constant of integration.
We can derive the kinematic equations for a constant acceleration using these integrals. With a(t) = a a constant, and doing the integration in (Figure), we find
v(t)=∫adt+C1=