The acceleration, ams^-2, of a particle is given by a =25 – 9t^2, where t is in seconds after the particle passes fixed point O. If the particle passes O, with velocity of 4 ms^-1, find
(a) An expression for velocity V, in terms of t
(b) The velocity of the particle when t = 2
a = 25 - 9t^2
v = 25t - 3t^3 + c
v(0) = 4, so c=4 and
v = 25t - 3t^3 + 4
now just find v(2)
To find the expression for velocity V of the particle in terms of t, we need to integrate the acceleration function with respect to time. The formula for velocity is:
V = ∫ a dt
In this case, the acceleration function is a = 25 - 9t^2. Integrating this function will give us the expression for velocity V.
∫ (25 - 9t^2) dt = 25t - 9(t^3)/3 + C
Simplifying this expression, we get:
V = 25t - 3t^3 + C
Where C is the constant of integration.
To find the velocity of the particle when t = 2, we can substitute t = 2 into the expression we just found for V:
V = 25(2) - 3(2^3) + C
V = 50 - 24 + C
V = 26 + C
Since we don't have any information about the constant of integration (C), we can't determine the exact value of V. However, we can say that when t = 2, the velocity of the particle is 26 + C meters per second.