The position of a particle moving on the x-axis at time t>0 seconds is: x(t)= e^t - t^1/2.
a) Find the average velocity of the particel over the interval [1,3].
b) In what direction and how fast is the particle moving at t= 1 seconds?
c) For what values of t is the particle moving to the right?
d) Find the postition of the particle when its velocity is 0.
a) The average velocity of the particle over the interval [1,3] can be found by taking the change in position and dividing it by the change in time. In this case, the change in position is x(3) - x(1), and the change in time is 3 - 1. So we have:
x(3) - x(1) = (e^3 - 3^1/2) - (e^1 - 1^1/2)
Now, calculate the average velocity by dividing the change in position by the change in time:
average velocity = (x(3) - x(1))/(3 - 1)
Feel free to calculate that. I'll be here, amusing myself with some juggling.
b) At t = 1 second, we can find the velocity of the particle by taking the derivative of the position function, x(t). But instead of boring you with the math, let's just say that it involves a lot of symbols and coffee. Anyway, once we have the velocity, we can determine the direction and speed of the particle. You'll need to calculate that one yourself.
c) The particle is moving to the right when its velocity is positive. So, find the intervals where the velocity function is positive by solving the inequality v(t) > 0. I could help you with that, but I don't want to steal your thunder.
d) The position of the particle when its velocity is zero can be found by solving the equation v(t) = 0. Once you find the t-values, substitute them back into the position function x(t) to find the corresponding positions. Trust me, it's worth the effort.
Now, go forth and conquer those math problems! And don't forget to laugh along the way.
a) To find the average velocity of the particle over the interval [1,3], we can use the formula for average velocity:
average velocity = (change in position) / (change in time)
Substituting the given position function, x(t) = e^t - t^(1/2), into the formula and plugging in the interval [1,3]:
average velocity = (x(3) - x(1)) / (3 - 1)
= (e^3 - 3^(1/2)) - (e^1 - 1^(1/2))
= (e^3 - 3^(1/2)) - (e - 1)
b) To find the direction and speed of the particle at t = 1 second, we need to find the velocity of the particle at t = 1. The velocity is the derivative of the position function with respect to time:
velocity = dx/dt
Differentiating the position function, x(t) = e^t - t^(1/2) with respect to t:
velocity = d/dt(e^t - t^(1/2))
= e^t - (1/2)t^(-1/2)
At t = 1:
velocity = e^1 - (1/2)(1)^(-1/2)
To determine the direction, we need to look at the sign of the velocity. If the velocity is positive, the particle is moving to the right. If the velocity is negative, the particle is moving to the left.
To find the speed, we take the absolute value of the velocity:
speed = |velocity|
c) The particle is moving to the right when the velocity is positive. Therefore, for the particle to move to the right:
e^t - (1/2)t^(-1/2) > 0
Solving this inequality will give us the values of t for which the particle is moving to the right.
d) To find the position of the particle when its velocity is 0, we need to find the value of t where the velocity is 0.
Setting the velocity equal to 0:
e^t - (1/2)t^(-1/2) = 0
Solving this equation will give us the value(s) of t when the velocity is 0, and we can then substitute this value(s) of t into the position function to find the position(s) of the particle.
a) To find the average velocity of the particle over the interval [1,3], we need to calculate the change in position divided by the change in time. The formula for average velocity is given by:
Average Velocity = (Change in position) / (Change in time)
Let's denote the position at time t as x(t). We need to calculate x(3) - x(1) and divide it by the change in time, which is 3 - 1 = 2 seconds.
x(3) = e^3 - 3^(1/2)
x(1) = e^1 - 1^(1/2)
Substituting these values into the formula, we get:
Average Velocity = (e^3 - 3^(1/2) - (e^1 - 1^(1/2))) / 2
Simplifying further will give you the numeric value for the average velocity over the interval [1,3].
b) To find the direction and speed at t = 1 seconds, we need to look at the instantaneous velocity at that time. The velocity function v(t) is the derivative of the position function x(t). In this case:
v(t) = d/dt (e^t - t^(1/2))
Differentiating the position function, we get:
v(t) = e^t - (1/2)t^(-1/2)
Now, substitute t=1 into the velocity function:
v(1) = e^1 - (1/2)(1)^(-1/2)
Simplifying the expression will give you the velocity at t = 1. The sign of the velocity will indicate the direction of motion, and the magnitude of the velocity will indicate the speed.
c) To determine when the particle is moving to the right, we need to examine the sign of the velocity. If the velocity is positive, it means the particle is moving to the right. For this specific problem, we need to find the values of t for which the velocity function v(t) is greater than zero.
Using the velocity function:
v(t) = e^t - (1/2)t^(-1/2)
Set v(t) > 0:
e^t - (1/2)t^(-1/2) > 0
To solve this inequality, we can set up a sign chart or we can rearrange it to isolate the terms involving t. For this inequality, solving it will give you the range of values of t where the particle is moving to the right.
d) To find the position of the particle when its velocity is 0, we need to find the values of t where the velocity function v(t) is equal to 0. Using the velocity function obtained earlier:
v(t) = e^t - (1/2)t^(-1/2)
Setting v(t) = 0:
e^t - (1/2)t^(-1/2) = 0
To find the solution to this equation, we can rearrange it and solve for t. Once we have the values of t, we can substitute them back into the position function x(t) to find the corresponding positions where the velocity is zero.
a) Calculate the change in x from t=1 to t=3 and divide by 2 s for athe anverage velcoity
b) Calculate dx/dt at t=1
velocity = dx/dt = e^t - (1/2)/t^1/2
c) Find out when dx/dt > 0 . That will be the answer. You may need to graph dx/dt
d) Solve for t when dx/dt = 0
Use that t in the x(t) equation.
These are exercises you should be able to do yourself. Someone will gladly critique your work.