If an arrow is shot upward on Mars with a speed of 52 m/s, its height in meters t seconds later is given by
y = 52t − 1.86t2.
(Round your answers to two decimal places.)
What is the average speed over the given time intervals?
(i) [1, 2]
(ii) [1, 1.5]
(iii) [1, 1.1]
(iv) [1, 1.01]
(v) [1, 1.001]
iii. answer is 48.094
48.094
Okay, let's calculate the average speeds for the given time intervals on Mars. Remember, we're talking about an arrow here, not a clown running around!
(i) [1, 2]:
To calculate the average speed, we need to find the change in height (Δy) and divide it by the change in time (Δt). For this interval:
Δy = y(2) - y(1) = (52(2) - 1.86(2)^2) - (52(1) - 1.86(1)^2)
Calculating this gives us Δy = 12.16 meters.
Δt = 2 - 1 = 1 second.
Now, we can calculate the average speed:
Average speed = Δy/Δt = 12.16/1 = 12.16 m/s.
(ii) [1, 1.5]:
Δy = y(1.5) - y(1) = (52(1.5) - 1.86(1.5)^2) - (52(1) - 1.86(1)^2)
Calculating this gives us Δy = 18.22 meters.
Δt = 1.5 - 1 = 0.5 seconds.
Average speed = Δy/Δt = 18.22/0.5 = 36.44 m/s.
(iii) [1, 1.1]:
Δy = y(1.1) - y(1) = (52(1.1) - 1.86(1.1)^2) - (52(1) - 1.86(1)^2)
Calculating this gives us Δy = 3.84 meters.
Δt = 1.1 - 1 = 0.1 seconds.
Average speed = Δy/Δt = 3.84/0.1 = 38.4 m/s.
(iv) [1, 1.01]:
Δy = y(1.01) - y(1) = (52(1.01) - 1.86(1.01)^2) - (52(1) - 1.86(1)^2)
Calculating this gives us Δy = 0.1924 meters.
Δt = 1.01 - 1 = 0.01 seconds.
Average speed = Δy/Δt = 0.1924/0.01 = 19.24 m/s.
(v) [1, 1.001]:
Δy = y(1.001) - y(1) = (52(1.001) - 1.86(1.001)^2) - (52(1) - 1.86(1)^2)
Calculating this gives us Δy = 0.019224 meters.
Δt = 1.001 - 1 = 0.001 seconds.
Average speed = Δy/Δt = 0.019224/0.001 = 19.224 m/s.
And there you have it! The average speeds for the given time intervals on Mars. Just remember, clown arrows tend to go a lot faster!
To find the average speed over a given time interval, we need to calculate the total distance traveled and divide it by the duration of the interval.
The general formula for average speed is:
Average Speed = (Total Distance Traveled) / (Duration)
Let's calculate the average speed for each given time interval:
(i) [1, 2]:
Duration = 2 - 1 = 1 second
To find the total distance traveled, we need to calculate the displacement. We can do this by finding the difference in heights at the two time points.
y(2) - y(1) = (52*2 - 1.86*2^2) - (52*1 - 1.86*1^2)
= (104 - 7.44) - (52 - 1.86)
= 96.56 - 50.14
= 46.42 meters
Average Speed = (Total Distance Traveled) / (Duration)
= 46.42 meters / 1 second
= 46.42 m/s
(ii) [1, 1.5]:
Duration = 1.5 - 1 = 0.5 seconds
Total Distance Traveled = y(1.5) - y(1)
= (52*1.5 - 1.86*1.5^2) - (52*1 - 1.86*1^2)
= (78 - 2.77) - (52 - 1.86)
= 75.23 - 50.14
= 25.09 meters
Average Speed = (Total Distance Traveled) / (Duration)
= 25.09 meters / 0.5 seconds
= 50.18 m/s
(iii) [1, 1.1]:
Duration = 1.1 - 1 = 0.1 seconds
Total Distance Traveled = y(1.1) - y(1)
= (52*1.1 - 1.86*1.1^2) - (52*1 - 1.86*1^2)
= (57.2 - 1.94) - (52 - 1.86)
= 55.26 - 50.14
= 5.12 meters
Average Speed = (Total Distance Traveled) / (Duration)
= 5.12 meters / 0.1 seconds
= 51.2 m/s
(iv) [1, 1.01]:
Duration = 1.01 - 1 = 0.01 seconds
Total Distance Traveled = y(1.01) - y(1)
= (52*1.01 - 1.86*1.01^2) - (52*1 - 1.86*1^2)
= (52.52 - 1.88) - (52 - 1.86)
= 50.64 - 50.14
= 0.50 meters
Average Speed = (Total Distance Traveled) / (Duration)
= 0.50 meters / 0.01 seconds
= 50.0 m/s
(v) [1, 1.001]:
Duration = 1.001 - 1 = 0.001 seconds
Total Distance Traveled = y(1.001) - y(1)
= (52*1.001 - 1.86*1.001^2) - (52*1 - 1.86*1^2)
= (52.052 - 1.8586) - (52 - 1.86)
= 50.1934 - 50.14
= 0.0534 meters
Average Speed = (Total Distance Traveled) / (Duration)
= 0.0534 meters / 0.001 seconds
= 53.4 m/s
Therefore, we have:
(i) [1, 2]: Average Speed = 46.42 m/s
(ii) [1, 1.5]: Average Speed = 50.18 m/s
(iii) [1, 1.1]: Average Speed = 51.2 m/s
(iv) [1, 1.01]: Average Speed = 50.0 m/s
(v) [1, 1.001]: Average Speed = 53.4 m/s
I will do one of them, you do the rest
(iii) [1, 1.1]
when x=1, y = 52-1.86 = 50.14
when x = 1.1, y = 52(1.1)-1.86(1.1)^2 = 54.72434
avg speed= (54.72434-50.14)/(1.1-1)
= 45.8434 m/s