The vector position of a particle varies in time according to the expression = (8.00 - 8.40t2 ) m. What is the velocity and acceleration as a function of time?
s = 8.00 - 8.40t^2 m
v = ds/dt = -16.8t m/s
a = dv/dt = -16.8 m/s^2
To find the velocity as a function of time, you need to take the derivative of the position vector with respect to time. Let's denote the position vector as r(t):
r(t) = (8.00 - 8.40t^2) m
To find the velocity, take the derivative of r(t) with respect to time, and we get:
v(t) = d(r(t))/dt
v(t) = d(8.00 - 8.40t^2)/dt
Differentiating the terms one by one:
v(t) = d(8.00)/dt - d(8.40t^2)/dt
Since the derivative of a constant is zero:
v(t) = 0 - 8.40 * d(t^2)/dt
Using the power rule, the derivative of t^2 is 2t:
v(t) = -8.40 * 2t
Simplifying further:
v(t) = -16.8t
Therefore, the velocity as a function of time is v(t) = -16.8t m/s.
To find the acceleration as a function of time, you need to take the derivative of the velocity vector with respect to time. Let's denote the velocity vector as v(t):
v(t) = -16.8t m/s
To find the acceleration, take the derivative of v(t) with respect to time, and we get:
a(t) = d(v(t))/dt
a(t) = d(-16.8t)/dt
Differentiating the term:
a(t) = -16.8 * d(t)/dt
Since the derivative of t is 1:
a(t) = -16.8 * 1
Simplifying further:
a(t) = -16.8
Therefore, the acceleration as a function of time is a(t) = -16.8 m/s^2.
To find the velocity and acceleration as a function of time, we need to differentiate the position vector with respect to time. Let's start with finding the velocity.
The position vector is given by r(t) = (8.00 - 8.40t^2) m
To find the velocity vector, we differentiate r(t) with respect to time (t):
v(t) = d/dt (8.00 - 8.40t^2) m
To differentiate the expression, we can use the power rule of differentiation.
For the constant term, 8.00, the derivative is zero because it does not depend on time.
For the term -8.40t^2, we apply the power rule: d/dt (ct^n) = nct^(n-1)
Applying the power rule, we get:
v(t) = -8.40 * 2t m/s
Simplifying further, the velocity as a function of time is:
v(t) = -16.80t m/s
Next, let's find the acceleration.
To find the acceleration vector, we differentiate the velocity vector with respect to time (t):
a(t) = d/dt (-16.80t) m/s
Again, applying the power rule, we get:
a(t) = -16.80 m/s^2
Therefore, the acceleration as a function of time is constant and equal to -16.80 m/s^2.