two particles are moving along a linear track according to the functions x1=t^2+2t and x2=-t^2+8t+80 for t is greater than or equal to zero.
a)at what value of t will the two particles be in the same place
b)will they be going in the same or opposite directions at the time they are in the same place?
c)at what value of t do they have the same velocity
d)during what time interval(s) are they moving in the same direction?
I need help with parts b and d
a)t^2+2t=-t^2+8t+80
2t^2+2t=8t+80
2t^2-6t-80=0
2(t^2-3t-40)=0
2(t-8)(t+5)=0
t=8 and t=-5, but we reject t=-5 because we cant have negative time
c) v1(t)=2t+2
v2(t)=-2t+8
2t+2=-2t+8
4t+2=8
4t=6
t=3/2
b) find the velocity of each when t = 8
v1 = 2(8)+2 = + 18
v2 = -2t +8 = -8
so in opposite directions
d ) they start out both moving +
when is v1-v2 = 0 ?
they both start out +
find out when the first one hits v = 0
then it goes - while the first is +
find when the second hits v=0
then they both are -
etc
oh, the first one is always v positive v = 2t+2
so when does the second one go - ?
b) To determine if the particles are moving in the same or opposite directions at the time they are in the same place, we can look at their velocities. The velocity of Particle 1, v1(t), is given by the derivative of its position function x1(t) with respect to time:
v1(t) = d/dt (x1(t))
v1(t) = d/dt (t^2 + 2t)
v1(t) = 2t + 2
Similarly, the velocity of Particle 2, v2(t), is given by the derivative of its position function x2(t) with respect to time:
v2(t) = d/dt (x2(t))
v2(t) = d/dt (-t^2 + 8t + 80)
v2(t) = -2t + 8
To determine if they are moving in the same or opposite directions, we need to evaluate the velocities at the time they are in the same place. We found earlier that t = 8 is the value at which the particles are in the same place:
v1(8) = 2(8) + 2 = 16 + 2 = 18
v2(8) = -2(8) + 8 = -16 + 8 = -8
Since v1(8) and v2(8) have opposite signs (positive and negative), the particles are moving in opposite directions at the time they are in the same place.
d) To determine the time interval during which the particles are moving in the same direction, we need to analyze their velocities. Recall that Particle 1 has a velocity given by v1(t) = 2t + 2, and Particle 2 has a velocity given by v2(t) = -2t + 8.
To have the same velocity, v1(t) must be equal to v2(t):
2t + 2 = -2t + 8
4t = 6
t = 6/4
t = 3/2
Therefore, the particles have the same velocity at t = 3/2.
To determine the time interval during which they are moving in the same direction, we need to analyze the signs of their velocities for different values of t. We found earlier that the particles are in the same place at t = 8.
For t < 3/2, both v1(t) and v2(t) are positive, indicating that they are moving in the same direction.
For t > 8, both v1(t) and v2(t) are negative, indicating that they are moving in the same direction.
Therefore, they are moving in the same direction during the time interval t < 3/2 and t > 8.