A football player attempts a field goal by kicking the football. The ball follows the path modelled by
the equation h=-4.9t2(means t squared)+10t+3, where h is the height of the ball above the ground
in metres, and t is the time sincethe ball was kicked in seconds.
1. Describe the path of the ball.
2. After how many seconds does the ball reach the ground?
3. The ball must clear the uprights for the field goal to count. Theuprights are approximately 5m
high. How long does the ball stay above 5m in height?
Solution:
(i) Describe the path of the ball.
Here we need to draw graph for the function h(t) = -4.9 t^2 +10t+3, where h is the height of the
ball.
t-is the independent variable and h is the dependent variable.
Plugin the values for t (0,1,2,3…) and find the corresponding values for h and draw the graph.
Here is the graph.
In the graph t represents x-axis and h represents y-axis.
(ii) After how many seconds does the ball reach the ground?
This means height of the ball is zero. That is h= 0
-4.9 t^2 + 10 t+ 3 = 0
This is nothing but quadratic equation. Here we need to use the quadratic formula to find the value
of t.
We know that the quadratic formula for ax^2+bx+c= 0 is –b +/- sqrt( b^2- 4ac)
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2a
In the given equation –4.9t^2+10t+3=0, we have a= -4.9 , b = 10 and c= 3
Plugin the value of a, b and c in the quadratic formula, we get
T= -10 +/- sqrt(100+58.8)
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-9.8
t= -10+/- 12.6
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[ sqrt(158.8)= 12.6 approximately]
-9.8
t= -10+12.6
_______
-9.8
so, t= -0.27
or t = -10 -12.6
_______
-9.8
or t= 2.3
Here time cannot be negative, so we can omit t= -0.27
Therefore the ball hits the ground after t = 2.3 seconds
Note: you can look at the graph also. Where the curve cuts at x-axis is know as the ball hits
the ground.
(iii) The ball must clear the uprights for the field goal to count. The uprights are approximately 5m
high. How long does the ball stay above 5m in height?
Look at the graph. The ball attains 5 m height , when t = 0.3 seconds
And again it returns the height 5 m after t= 1.9 seconds (please refer the graph)
We need to find how long the ball stay above 5 m height.
Required time T1 = 1.9 – 0.3
T1 = 1.6 seconds
The ball stay 1.6 seconds above the 5m in height.
That’s it
Hope you can understand it.