A particle moves at varying velocity along a line and s=f(t) represents the particle's distance from a point as a function of time, t. Sketch a possible graph for f if the average velocity of the particle between t=2 and
t=6 is the same as the instantaneous velocity at t=5.
I'm confused on how I know what to graph without any points. How do I find points?
well one possible curve is a horizontal line, s = f(t) = f(5)
To determine the graph of the function f(t) that represents the particle's distance from a point as a function of time, it is helpful to consider the relationship between velocity and distance.
Velocity is the rate of change of distance with respect to time. Mathematically, it is given by the derivative of the distance function with respect to time: v(t) = f'(t), where v(t) represents the instantaneous velocity at time t and f'(t) is the derivative of f(t).
Average velocity, on the other hand, is the total change in distance divided by the total change in time. For any interval [a, b], the average velocity is calculated as:
average velocity = (f(b) - f(a)) / (b - a)
In this case, the problem states that the average velocity between t=2 and t=6 is the same as the instantaneous velocity at t=5. Let's denote the instantaneous velocity at t=5 as v(5).
Since average velocity and instantaneous velocity are equal, we can equate the two:
(f(6) - f(2)) / (6 - 2) = v(5)
This equation allows us to find possible values for f(2), f(6), and v(5).
For example, you could arbitrarily choose a value for v(5), say v(5) = 3. Then, the equation becomes:
(f(6) - f(2)) / 4 = 3
By rearranging this equation, we can express it as:
f(6) - f(2) = 12
Now, you have a constraint on the possible values of f(6) and f(2): their difference must be 12 for the average velocity and instantaneous velocity to be equal.
Given this constraint, you can choose any values for f(2) and f(6) that satisfy the equation f(6) - f(2) = 12. For instance, let's say you pick f(2) = 4, then f(6) must be 16, as 16 - 4 = 12.
With these chosen values, you have two points: (2, 4) and (6, 16). Plotting these points on a graph will allow you to sketch a potential graph for f(t).
Remember, this is just one example. You can choose different values for v(5), f(2), and f(6) and repeat the process to obtain various possible graphs for f(t) that satisfy the given conditions.