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Please solve and show graph using Excel if needed.
1. Maximum height. If a baseball is projected upward from ground level with an
initial velocity of 64 feet per second, then its height is a function of time,
given by
s(t) = -16t^2 + 64t. Graph this function for 0 < t < 4.
What is the maximum height reached by the ball?
Solution:
Before solving this question I would like to remind you that whenever anything is thrown
upwards the earth exerts a force to pull back the object back to the ground.
Due to the force of earth the object constantly decelerates with the speed of 9.8m/s2,this
acceleration is due to force of gravity of earth. At the highest point the velocity of object
thrown becomes zero and the object tends to come back to earth.
Any how that was just brief physics behind that.
Now coming to the main problem
The object was thrown at the speed of 64 feet per second
Hence initial speed of the object u=64feet/s
We are given that the height of the object at any time is given by s(t) = -16t^2 + 64t
Taking first differential of the above expression to calculate speed we have
ds
=-32t+64
dt
Hence velocity at any point = -32t+64 feet/se
As at the highest point the velocity becomes zero so putting it equal to zero we have
0= -32t+64 feet/se
or -32t=-64
or t=64/32
or t=2 seconds
Hence the speed of the object will become zero after 2 seconds or at the highest point as
we have earlier discussed in the starting.
Putting value of t =2 in the given expression of s we have
s(t) = -16t^2 + 64t
s=-16(2)2 +64*2
s=-64+128
s=64 feets
Hence the highest point reached by the objsct is 64 feets
Part 2
Graphing of the height function for 0<t<4
Time t
Height = s(t) = -16t^2 + 64t
0
1
2
3
4
0
48
64
48
0
We can plot the graph for the above function easily using excel
Height and Time Variation
70
64
60
Height
50
48
48
40
30
20
10
0
0
0
0
1
2
3
4
Tim e
Now I would again like to have brief discussion about the graph,
What we are having is that at initial point when we through the object up it starts to rise
upwards with velocity of 64 f/s
Slowly and slowly the speed decreases and the object comes to halt at highest point,
After that the object again starts to descend down and the speed increase and the speed
ultimately becomes zero at the lowest point
Hope the solutions is clear to you, please feel free to contact us again in case of doubt