measured in meters/second. The acceleration is the derivative of the velocity, or
) = –9.8
This constant acceleration is due to the Earth's gravity and is measured in meters/second². It is negative because the acceleration is downward. In the applet, the left-hand graph shows the object as a black dot on a vertical line. The right-hand graphs plots s
(in magenta), v
(in blue) and a
(in red), all against t
. Click the Start button to run the animation. Note that the object goes straight up and then down, because in this problem the object is just moving vertically. It is easy to get confused and think of the magenta curve as showing the path of the object, but this is not true; the magenta curve is a plot of height vs. time.
In this example, the object reaches a peak and then falls back to Earth, eventually hitting the ground. How can we find the time when these two events happen? To find when the object hits the ground, we want to find when s(t) = 0. In other words, we want to solve
0 = -4.9t
² + 20t
One solution is obviously t
= 0, because the object started out at ground level. The other can be found using algebra and is t
≈ 4.082. To find when the object reaches its peak, we want to find when the velocity is zero, which means we want to solve 0 = -9.8t
+ 20. A little algebra yields the answer t
2. More complex motion
Select the second example. This shows a more complex motion over the time interval 0 ≤ t ≤ 5. Click Start to watch the animation of the black dot moving vertically. How would you find out when the object is at s = 0? How would you find out when the object reverses direction?
3. Oscillating motions
Select the third example, showing oscillating motion on the interval 0 ≤ t ≤ 6π. Click Start to run the animation. How could you find out when the object reaches one of its highest points? Lowest points? Maximum velocity? Maximum acceleration?
Try your own examples. Enter a function using t as the variable. Set the values of tmin, tmax, smin, and smax (or pan and zoom using the mouse). The animation and the slider will run from tmin to tmax.
Other 'Applications of Differentiation' topics
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