If a ball is thrown straight up into the air with an initial velocity of 95 ft/s, it height in feet after t second is given by y=95t–16t2. Find the average velocity for the time period begining when t=1 and lasting
(i) 01 seconds:
(ii) 001 seconds:
(iii) 0001 seconds:
Finally based on the above results, guess what the instantaneous velocity of the ball is when t=1.
When a ball is thrown up into the air, it makes the shape of a parabola. The equation S= -16t^2 + v*t + k gives the height of the ball at any time, t in seconds, where “v” is the init
Oh, I see you're getting into some physics calculations! Well, I may not be the most mathematically inclined clown, but I'll give it a shot!
(i) To find the average velocity for the time period from t = 1 to t = 1.01 seconds, we can plug these values into the equation y = 95t - 16t^2.
When t = 1, we have y = 95(1) - 16(1)^2 = 79 feet.
When t = 1.01, we have y = 95(1.01) - 16(1.01)^2 = 79.2406 feet.
The average velocity is given by (change in height)/(change in time). So, the average velocity is (79.2406 - 79)/(1.01 - 1) = 24.06 ft/s.
(ii) Similarly, let's calculate the average velocity for the time period from t = 1 to t = 1.001 seconds:
When t = 1, we have y = 79 feet.
When t = 1.001, we have y = 78.924 feet.
Average velocity = (78.924 - 79)/(1.001 - 1) = -7.6 ft/s.
(iii) Now, for the time period from t = 1 to t = 1.0001 seconds:
When t = 1, y = 79 feet.
When t = 1.0001, y = 78.9924 feet.
Average velocity = (78.9924 - 79)/(1.0001 - 1) = -0.76 ft/s.
Based on these calculations, it seems like the instantaneous velocity at t = 1 is decreasing! So, I would guess that the instantaneous velocity of the ball at t = 1 is a negative value since it's going upward and slowing down. But remember, this is just a guess based on the average velocities. The actual instantaneous velocity would require some fancy calculus!
To find the average velocity, we need to calculate the change in height divided by the change in time.
(i) For a time period of 01 seconds:
To find the average velocity during this time, we need to evaluate the expression y=95t-16t^2 for t=1 and t=1+0.1.
1. Substitute t=1 into the equation:
y(1) = 95(1) - 16(1)^2
y(1) = 95 - 16 = 79 ft
2. Substitute t=1.1 into the equation:
y(1.1) = 95(1.1) - 16(1.1)^2
y(1.1) = 104.5 - 19.36 = 85.14 ft
The change in height is 85.14 - 79 = 6.14 ft.
The change in time is 0.1 seconds.
Average velocity = change in height / change in time
Average velocity = 6.14 ft / 0.1 s
Average velocity ≈ 61.4 ft/s
(ii) For a time period of 001 seconds:
To find the average velocity during this time, we need to evaluate the expression y=95t-16t^2 for t=1 and t=1+0.01.
1. Substitute t=1 into the equation:
y(1) = 95(1) - 16(1)^2
y(1) = 95 - 16 = 79 ft
2. Substitute t=1.01 into the equation:
y(1.01) = 95(1.01) - 16(1.01)^2
y(1.01) = 95.95 - 16.2416 = 79.7084 ft
The change in height is 79.7084 - 79 = 0.7084 ft.
The change in time is 0.01 seconds.
Average velocity = change in height / change in time
Average velocity = 0.7084 ft / 0.01 s
Average velocity ≈ 70.84 ft/s
(iii) For a time period of 0001 seconds:
To find the average velocity during this time, we need to evaluate the expression y=95t-16t^2 for t=1 and t=1+0.001.
1. Substitute t=1 into the equation:
y(1) = 95(1) - 16(1)^2
y(1) = 95 - 16 = 79 ft
2. Substitute t=1.001 into the equation:
y(1.001) = 95(1.001) - 16(1.001)^2
y(1.001) = 95.095 - 16.032032001 = 78.062968 ft
The change in height is 78.062968 - 79 = -0.937032 ft.
The change in time is 0.001 seconds.
Average velocity = change in height / change in time
Average velocity = -0.937032 ft / 0.001 s
Average velocity ≈ -937.032 ft/s
Based on the above results, we can observe that as the time period decreases, the average velocity approaches the actual instantaneous velocity at t=1. Therefore, we can estimate that the instantaneous velocity of the ball when t=1 is approximately 937.032 ft/s in the downward direction.
i)
when t=1, y = 95-16 = 79
when t=1.01 , y = 95(1.01) - 16(1.01)^2 = 79.6284
avg vel = (79.6284-79)/(1.01-1) = 62.84
repeat for ii) and iii)
you should be able to draw your conclusion by looking at those results.