The position of an object as a function of time is given as x = At3 + Bt2 + Ct + D. The constants
are A = 2.1 m/s3
, B = 1.0 m/s2
, C = –4.1 m/s, and D = 3 m.
a) What is the velocity of the object at t = 10.0 s?
b) At what time(s) is the object at rest?
c) What is the acceleration of the object at t = 0.50 s?
d) Plot the acceleration as a function of time for the time interval from t = –10.0 s to t =
10.0 s.
To answer these questions, we need to take derivatives of the given position function with respect to time. Each derivative represents a different physical quantity.
Let's go through each question step by step:
a) To find the velocity of the object at t = 10.0 s, we need to find the derivative of the position function with respect to time. The derivative of x with respect to t represents velocity.
Taking the derivative of the position function x = At^3 + Bt^2 + Ct + D with respect to t, we get:
v = dx/dt = 3At^2 + 2Bt + C
Substituting the values of A, B, and C, we get:
v = 3(2.1 m/s^3)(10.0 s)^2 + 2(1.0 m/s^2)(10.0 s) + (-4.1 m/s)
Evaluating this expression gives us the velocity at t = 10.0 s.
b) To find when the object is at rest, we need to find the time(s) when the velocity is equal to zero. We can set the velocity equation equal to zero and solve for t:
3At^2 + 2Bt + C = 0
Using the given values of A, B, and C, we can solve this quadratic equation to find the time(s) when the object is at rest.
c) To find the acceleration of the object at t = 0.50 s, we need to take the derivative of the velocity function with respect to time. The second derivative of x represents acceleration.
Taking the derivative of the velocity function v = 3At^2 + 2Bt + C with respect to t, we get:
a = dv/dt = 6At + 2B
Substituting the values of A and B, we can evaluate this expression to find the acceleration at t = 0.50 s.
d) Finally, to plot the acceleration as a function of time from t = -10.0 s to t = 10.0 s, we can use the acceleration equation we derived in part c. Substitute values of t from -10.0 s to 10.0 s into the equation and plot the corresponding values of acceleration on a graph with time on the x-axis and acceleration on the y-axis.