The equation y=-4.9t^2+3.5t+5 describes the height (in meters) of a ball thrown upward at 3.5 meters per second from 5 meters above the ground. In how many seconds will the ball hit the ground? Express your answer as a common fraction.
Thank you for your help!
the ball hits the ground when y=0.
If you want a fraction, then let's get rid of the decimals:
y = 1/10 (-49t^2 + 35t + 50)
= 1/10 (7t+5)(10-7t)
y=0 when t = 10/7
To find the time at which the ball hits the ground, we need to find the value of t when the height (y) is equal to zero.
We have the equation y = -4.9t^2 + 3.5t + 5.
Set y to zero: 0 = -4.9t^2 + 3.5t + 5.
Now we need to solve this quadratic equation for t.
Multiply every term by -1 to make the quadratic equation positive: 0 = 4.9t^2 - 3.5t - 5.
Apply the quadratic formula: t = (-b ± √(b^2 - 4ac)) / 2a.
In this equation, a = 4.9, b = -3.5, and c = -5.
Substitute these values into the quadratic formula: t = (-(-3.5) ± √((-3.5)^2 - 4(4.9)(-5))) / (2(4.9)).
Simplify the equation: t = (3.5 ± √(12.25 + 98)) / 9.8.
Simplify further: t = (3.5 ± √(110.25)) / 9.8.
Now, we need to find the two possible values of t by evaluating this equation with both the plus and minus signs:
t1 = (3.5 + √(110.25)) / 9.8, and t2 = (3.5 - √(110.25)) / 9.8.
Calculating the values gives t1 ≈ 1.166 seconds, and t2 ≈ -0.469 seconds.
Since time cannot be negative in this context, we discard the negative value.
Therefore, the ball will hit the ground approximately 1.166 seconds after it is thrown.
To find the time it takes for the ball to hit the ground, we need to determine when the height (y) is equal to zero. Let's set y = 0 in the equation:
0 = -4.9t^2 + 3.5t + 5
Now, we have a quadratic equation. We can solve it using the quadratic formula, which is:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -4.9, b = 3.5, and c = 5. Plugging these values into the quadratic formula, we get:
t = (-(3.5) ± √((3.5)^2 - 4(-4.9)(5))) / (2(-4.9))
Simplifying further:
t = (-3.5 ± √(12.25 + 98)) / (-9.8)
t = (-3.5 ± √(110.25)) / (-9.8)
Calculating the square root:
t = (-3.5 ± 10.5) / (-9.8)
Now, we have two possible solutions:
1) t = (-3.5 + 10.5) / (-9.8)
2) t = (-3.5 - 10.5) / (-9.8)
1) t = 7 / -9.8 = -0.7143
2) t = -14 / -9.8 = 1.4286
Since time cannot be negative in this context, we discard the negative value. Therefore, the ball will hit the ground in approximately 1.4286 seconds.
Answer: 1.4286 seconds (approximately) or 4/7 seconds (exact answer)