Suppose that an object moves along an s-axis so that its location is given by s(t)=t^2+4t at time t.(Here is in meters and is in seconds.)
(a) Find the average velocity of the object in meters per second over the time interval t=2 to t=9 seconds.
Average velocity = ? m/s.
(b) Find the instantaneous velocity of the object in meters per second at t=5 seconds.
Instantaneous velocity = ? m/s.
same way as the other one but here
v = ds/dt = 2 t + 4
so at t = 5
v = 2(5) + 4 = 14 m/s
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To find the average velocity of the object over the time interval t=2 to t=9 seconds, we use the formula:
Average velocity = (change in position) / (change in time)
1. First, we find the position of the object at t=2 seconds and t=9 seconds.
- At t=2 seconds: s(2) = (2)^2 + 4(2) = 4 + 8 = 12 meters
- At t=9 seconds: s(9) = (9)^2 + 4(9) = 81 + 36 = 117 meters
2. Next, we calculate the change in position and change in time.
- Change in position = s(9) - s(2) = 117 - 12 = 105 meters
- Change in time = 9 - 2 = 7 seconds
3. Finally, we plug these values into the formula for average velocity:
Average velocity = (105 meters) / (7 seconds) = 15 meters per second
So, the average velocity of the object over the time interval t=2 to t=9 seconds is 15 m/s.
To find the instantaneous velocity of the object at t=5 seconds, we take the derivative of the position function s(t) with respect to time t.
Position function: s(t) = t^2 + 4t
1. Take the derivative of the position function with respect to time t:
s'(t) = 2t + 4
2. Substitute t=5 into the derivative function:
s'(5) = 2(5) + 4 = 10 + 4 = 14 meters per second
So, the instantaneous velocity of the object at t=5 seconds is 14 m/s.