The position of a particular particle as a function of time is given by r = ( 9.60ti + 8.85j - 1.00t^2 k)m, where t is in seconds.
a) Determine the particle's velocity as a function of time.
b) Determine the particle's acceleration as a function of time.
(Express both answers in terms of the unit vectors i, j, and k)
v = dr/dt, so just differentiate each component with respect to t
v=9.6 i - 2.00t k
a=dv/dt = -2.00 k
Well, it is correct as long as the coordinate system is not changing.
To find the particle's velocity as a function of time, we need to take the derivative of the position vector with respect to time.
Given: r = (9.60ti + 8.85j - 1.00t^2 k)m
a) Velocity (v) as a function of time:
To find the velocity vector, we differentiate each component of the position vector with respect to time.
Differentiating r with respect to t:
dr/dt = (9.60i + 8.85j - 2.00tk)m/s
Therefore, the particle's velocity as a function of time is:
v = 9.60i + 8.85j - 2.00tk m/s
b) Acceleration (a) as a function of time:
To find the acceleration vector, we differentiate the velocity vector with respect to time.
Differentiating v with respect to t:
dv/dt = -2.00k m/s^2
Therefore, the particle's acceleration as a function of time is:
a = -2.00k m/s^2
To determine the particle's velocity as a function of time, we need to take the derivative of the position function with respect to time. Similarly, to determine the particle's acceleration as a function of time, we need to take the derivative of the velocity function with respect to time.
Let's start with finding the velocity function.
a) Velocity as a function of time:
The position function is given by r = (9.60ti + 8.85j - 1.00t^2k)m.
Take the derivative of each component of the position function with respect to time (t):
dr/dt = d/dt (9.60ti + 8.85j - 1.00t^2k)m
The derivative of the position function component by component is as follows:
1) The derivative of 9.60ti with respect to t is 9.60i.
2) The derivative of 8.85j with respect to t is 0 (as it doesn't depend on t).
3) The derivative of -1.00t^2k with respect to t is -2.00tk.
So, the velocity function is:
v = dr/dt = 9.60i - 2.00tk
Therefore, the particle's velocity as a function of time is given by v = 9.60i - 2.00tk.
b) Acceleration as a function of time:
Now, to find the particle's acceleration as a function of time, we need to take the derivative of the velocity function, v, with respect to time (t).
dv/dt = d/dt(9.60i - 2.00tk)
The derivative of 9.60i with respect to t is 0 since the vector i doesn't change with time.
The derivative of -2.00tk with respect to t is -2.00k since the t is treated as a constant in this derivative.
Therefore, the acceleration function is:
a = dv/dt = -2.00k
The particle's acceleration as a function of time is given by a = -2.00k.
Thus, the answers are:
a) Velocity function: v = 9.60i - 2.00tk
b) Acceleration function: a = -2.00k