If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y = 10t - 1.86t^2

a) Find the average velocity over the given time in intervals:

(i) [1,2] (ii) [1,1.5] (iii) [1,1.1] (iv) [1,1.01] (v) [1,1.001]

Do I just plug each in like this?

y= 10(1) - 1.86(2)^2
y= 2.56 m/s

OR do I use this: s(t)= 4.9t^2 like:

= [4.9(2)^2 - 4.9(1)^2]/1
= 14.7 m/s

I am not sure which way is correct or if either way is correct? Please help and explain too.

the purpose of the question is to notice that your answer will approach a certain fixed value as the interval gets shorter and shorter.

I will do the third one, you do the rest

[1,1.1]

y1 = 10(1) - 1.86(1)^2 = 8.14
y2 = 10(1.1) - 1.86(1.1)^2 = 8.7494

average velocity over given interval = (8.7494-8.14)/(1.1-1) = 6.094

Well, you're on the right track, but it seems like you may be mixing up a few things. Let's clarify!

The formula given for height y is correct: y = 10t - 1.86t^2. This equation represents the height of the rock at time t.

Now let's tackle the average velocity over the given time intervals. Average velocity is calculated by finding the change in position (in this case the change in height) divided by the change in time. The formula for average velocity is:

Average velocity = (final height - initial height) / (final time - initial time)

For example, for the interval [1, 2], the initial time is 1 and the final time is 2. Plugging these values into the formula, we have:

Average velocity = (y(2) - y(1)) / (2 - 1)

Here, y(1) represents the height of the rock at time t=1, so substitute t=1 into the y equation:

y(1) = 10(1) - 1.86(1)^2
y(1) = 8.14 meters

Similarly, you can calculate y(2) to find the height at time t=2:

y(2) = 10(2) - 1.86(2)^2
y(2) = 16.4 meters

Now, we can finally calculate the average velocity:

Average velocity = (y(2) - y(1)) / (2 - 1)
Average velocity = (16.4 - 8.14) / 1
Average velocity = 8.26 m/s

Repeat this process for the other time intervals [1, 1.5], [1, 1.1], [1, 1.01], and [1, 1.001] to find their respective average velocities.

Remember, the formula s(t) = 4.9t^2 you mentioned is for calculating the height of an object in free fall, not for this particular scenario. So stick with the given formula y = 10t - 1.86t^2.

Hope that clears things up!

To find the average velocity over a given time interval, you need to calculate the change in displacement divided by the change in time. Here's how you can solve each part:

(i) [1, 2]:
To find the average velocity for this interval, you need to calculate the change in displacement from t = 1 to t = 2. Plug in these values in the equation:
y = 10t - 1.86t^2

When t = 1, y = 10(1) - 1.86(1)^2 = 8.14 m
When t = 2, y = 10(2) - 1.86(2)^2 = 10.24 m

The change in displacement is 10.24 m - 8.14 m = 2.1 m.
The change in time is 2 s - 1 s = 1 s.

Average velocity = (Change in displacement) / (Change in time) = 2.1 m / 1 s = 2.1 m/s.

(ii) [1, 1.5]:
Repeat the same process for this interval.
When t = 1, y = 10(1) - 1.86(1)^2 = 8.14 m
When t = 1.5, y = 10(1.5) - 1.86(1.5)^2 = 9.41 m

The change in displacement is 9.41 m - 8.14 m = 1.27 m.
The change in time is 1.5 s - 1 s = 0.5 s.

Average velocity = (Change in displacement) / (Change in time) = 1.27 m / 0.5 s = 2.54 m/s.

(iii) [1, 1.1]:
When t = 1, y = 8.14 m.
When t = 1.1, y = 10(1.1) - 1.86(1.1)^2 = 8.37 m.

The change in displacement is 8.37 m - 8.14 m = 0.23 m.
The change in time is 1.1 s - 1 s = 0.1 s.

Average velocity = (Change in displacement) / (Change in time) = 0.23 m / 0.1 s = 2.3 m/s.

(iv) [1, 1.01]:
When t = 1, y = 8.14 m.
When t = 1.01, y = 10(1.01) - 1.86(1.01)^2 = 8.1547 m.

The change in displacement is 8.1547 m - 8.14 m = 0.0147 m.
The change in time is 1.01 s - 1 s = 0.01 s.

Average velocity = (Change in displacement) / (Change in time) = 0.0147 m / 0.01 s = 1.47 m/s.

(v) [1, 1.001]:
When t = 1, y = 8.14 m.
When t = 1.001, y = 10(1.001) - 1.86(1.001)^2 = 8.14147 m.

The change in displacement is 8.14147 m - 8.14 m = 0.00147 m.
The change in time is 1.001 s - 1 s = 0.001 s.

Average velocity = (Change in displacement) / (Change in time) = 0.00147 m / 0.001 s = 1.47 m/s.

So, the correct answer for each interval is:

(i) [1,2]: 2.1 m/s
(ii) [1,1.5]: 2.54 m/s
(iii) [1,1.1]: 2.3 m/s
(iv) [1,1.01]: 1.47 m/s
(v) [1,1.001]: 1.47 m/s

To find the average velocity over a given time interval, you need to calculate the change in height (∆y) divided by the change in time (∆t) within that interval.

In this case, the equation given for the height of the rock is y = 10t - 1.86t^2.

To find the average velocity over the interval [1,2], you would use the equation:

Average velocity = (∆y) / (∆t) = (y2 - y1) / (t2 - t1).

Substituting the values into the equation:

y1 = 10(1) - 1.86(1)^2 = 10 - 1.86 = 8.14 m
y2 = 10(2) - 1.86(2)^2 = 20 - 7.44 = 12.56 m
t1 = 1
t2 = 2

∆y = y2 - y1 = 12.56 - 8.14 = 4.42 m
∆t = t2 - t1 = 2 - 1 = 1 s

Average velocity = (∆y) / (∆t) = 4.42 / 1 = 4.42 m/s

For the other intervals, you would follow the same procedure.

(i) [1,2]:
∆y = y2 - y1 = 12.56 - 8.14 = 4.42 m
∆t = t2 - t1 = 2 - 1 = 1 s

Average velocity = (∆y) / (∆t) = 4.42 / 1 = 4.42 m/s

(ii) [1,1.5]:
∆y = y1.5 - y1 = (10(1.5) - 1.86(1.5)^2) - (10(1) - 1.86(1)^2)
∆y = 12.15 - 8.14 = 4.01 m
∆t = 1.5 - 1 = 0.5 s

Average velocity = (∆y) / (∆t) = 4.01 / 0.5 = 8.02 m/s

(iii) [1,1.1]:
∆y = y1.1 - y1 = (10(1.1) - 1.86(1.1)^2) - (10(1) - 1.86(1)^2)
∆y = 10.6 - 8.14 = 2.46 m
∆t = 1.1 - 1 = 0.1 s

Average velocity = (∆y) / (∆t) = 2.46 / 0.1 = 24.6 m/s

(iv) [1,1.01]:
∆y = y1.01 - y1 = (10(1.01) - 1.86(1.01)^2) - (10(1) - 1.86(1)^2)
∆y = 10.1 - 8.14 = 1.96 m
∆t = 1.01 - 1 = 0.01 s

Average velocity = (∆y) / (∆t) = 1.96 / 0.01 = 196 m/s

(v) [1,1.001]:
∆y = y1.001 - y1 = (10(1.001) - 1.86(1.001)^2) - (10(1) - 1.86(1)^2)
∆y = 10.01 - 8.14 = 1.87 m
∆t = 1.001 - 1 = 0.001 s

Average velocity = (∆y) / (∆t) = 1.87 / 0.001 = 1870 m/s

So the correct values for average velocities for each interval are:

(i) [1,2]: 4.42 m/s
(ii) [1,1.5]: 8.02 m/s
(iii) [1,1.1]: 24.6 m/s
(iv) [1,1.01]: 196 m/s
(v) [1,1.001]: 1870 m/s

The values you calculated using the s(t) = 4.9t^2 equation are incorrect because that equation does not represent the height of the rock over time in this scenario. Always make sure to use the correct equation given in the problem.