The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion
s = 2 sin(πt) + 2 cos(πt),
where t is measured in seconds. (Round your answers to two decimal places.)
(a) Find the average velocity during each time period.
(a) you give no time periods, but the average velocity over [a,b] is just the total distance divided by the total time:
(s(b)-s(a))/(b-a)
(i) [1, 2]
cm/s
(ii) [1, 1.1]
cm/s
(iii) [1, 1.01]
cm/s
(iv) [1, 1.001]
cm/s
(b) Estimate the instantaneous velocity of the particle when t = 1.
cm/s
oobleck told you how to do the average
(position at end - position at beginning) / time elapsed
for instantaneous, differentiate
ds/dt = velocity = d/dt[ 2 sin(πt) + 2 cos(πt) ]
v(t) = 2 pi cos ( pi t ) - 2 pi sin (pi t)
if t = 1
v = 2 pi cos pi - 2 pi sin pi
= 2 pi (-1) - 0
= -2 pi
By the way they luckily asked for displacement and velocity, not average distance and speed. If they asked for the scalars and not the vectors you would have to be careful about going back and forth. If you go all the way around a circular track your displacement and average velocity is ZERO. However your distance is pi D and speed is pi D/time
This problem is trying to get you to converge to the speed at t = 1
which we know is -2 pi
so try the part iii for example
s(1.01 ) = 2 sin(1.01 pi ) + 2 cos(1.01 pi)
= -0.0628 -1.999 = -2.062
s(1) = 2(0)+2(-1) = -2
so
s(1.01)-s(1) = -.062
divide by time of 0.01 seconds
v average = -6.1813
the exact from the second part was -2 pi = - -6.2318 .......
if you did 1 to 1.0001 seconds it would be even closer
To find the average velocity during each time period, we need to calculate the change in displacement over the given time period and divide it by the time interval.
In this case, we're given the equation of motion for the displacement of the particle, which is:
s = 2 sin(πt) + 2 cos(πt)
To find the average velocity during a specific time period, we need to find the change in displacement (Δs) over that time period. In other words, we need to find the difference in the values of s at the beginning and the end of the time period.
Let's consider a specific time period, say from t = a to t = b. The average velocity can be calculated as:
Average Velocity = (Δs) / (Δt)
Where:
Δs = s(b) - s(a) (change in displacement)
Δt = b - a (change in time)
We need to find the change in displacement (Δs) and divide it by the change in time (Δt) to get the average velocity.
In this case, the equation of motion is given by:
s = 2 sin(πt) + 2 cos(πt)
To find the average velocity during each time period, we need to take the derivative of s with respect to t, which will give us the velocity function. Then, we can evaluate the velocity function at the beginning and the end of each time interval to find the change in displacement.
Let's calculate the average velocity for a specific time period, say from t = a to t = b:
1. Take the derivative of the equation of motion to find the velocity function:
v = ds/dt
To find the derivative of the given equation, we apply the chain rule. The derivative of sin(πt) with respect to t is πcos(πt), and the derivative of cos(πt) with respect to t is -πsin(πt). Therefore, the velocity function (v) is:
v = (d/dt) (2sin(πt) + 2cos(πt))
= 2(πcos(πt)) - 2(πsin(πt))
= 2π(cos(πt) - sin(πt))
2. Evaluate the velocity function (v) at t = a and t = b:
v(a) = 2π(cos(πa) - sin(πa))
v(b) = 2π(cos(πb) - sin(πb))
3. Find the change in displacement (Δs) by subtracting the values of s at the beginning and the end of the time period:
Δs = s(b) - s(a)
= (2sin(πb) + 2cos(πb)) - (2sin(πa) + 2cos(πa))
4. Calculate the change in time (Δt) by subtracting the values of t at the beginning and the end of the time period:
Δt = b - a
5. Finally, calculate the average velocity by dividing the change in displacement (Δs) by the change in time (Δt):
Average Velocity = (Δs) / (Δt)
Repeat this process for each time period to find the average velocity during that period. Remember to round your answers to two decimal places.