A particle moves along the x axis. Its position is given by the equation x = 2.00 + 3.20t - 3.85 t2 with x in meters and t in seconds. Determine its position at the instant it changes direction. (in m)
x = 2 + 3.2 t - 3.85 t^2
when is v = 0?
dx/dt = v = 3.2 - 7.7 t
v = 0 when t = 3.2/7.7 = .416 seconds
then
x = 2 + 1.33 - .665 = 2.66
If you do not know calculus, find the vertex of the original equation.
To determine the position at the instant the particle changes direction, we need to find the value of t when the velocity of the particle is zero.
The velocity of the particle can be obtained by taking the derivative of the equation for position with respect to time, t.
Differentiating the equation x = 2.00 + 3.20t - 3.85t^2:
dx/dt = 3.20 - 2 * 3.85t
Setting the velocity equal to zero:
0 = 3.20 - 2 * 3.85t
Rearranging the equation:
6.40t = 3.20
Divide both sides by 6.40:
t = 3.20 / 6.40
t = 0.50 seconds
To find the position at this instant, substitute this value of t back into the equation for x:
x = 2.00 + 3.20(0.50) - 3.85(0.50)^2
x = 2.00 + 1.60 - 3.85(0.25)
x = 2.00 + 1.60 - 0.96
x = 2.64 meters
Therefore, at the instant the particle changes direction, its position is 2.64 meters.