A ball is thrown vertically upwards From the top of a building of height 29.4 m and with an initial velocity 24.5 m/sec. If the height H of the ball from the ground level is given by H = 29.4 + 24.5t - 4.9t², then find the time taken by the ball to reach the ground
4.9 t^2 -24.5 t -29.4 = 0
t^2 - 5 t - 6 = 0
(t-6)(t+1) = 0
t = 6 seconds (the -1 second is before release :)
Well, let's walk through this step by step. First, we know that the height of the ball from the ground level is given by H = 29.4 + 24.5t - 4.9t².
Now, to find the time taken by the ball to reach the ground, we need to figure out when the height (H) is equal to zero. In other words, we need to solve the equation 29.4 + 24.5t - 4.9t² = 0.
To solve this quadratic equation, we can use the quadratic formula: t = (-b ± √(b² - 4ac)) / (2a).
Plugging in the values for a, b, and c, we get: t = (-(24.5) ± √((24.5)² - 4(-4.9)(29.4))) / (2(-4.9)).
Simplifying further, we have: t = (-24.5 ± √(600.25 + 573.12)) / (-9.8).
Calculating the square root and adding the values, we get: t = (-24.5 ± √(1173.37)) / (-9.8).
Now, since the ball is thrown vertically upwards, we only consider the positive value for t, as negative time does not make sense in this context.
So, t = (-24.5 + √(1173.37)) / (-9.8).
Using a calculator, we find t ≈ 2.365 seconds.
Therefore, the time taken by the ball to reach the ground is approximately 2.365 seconds.
But hey, keep in mind that we're assuming no other factors like air resistance, so this is a theoretical calculation. In reality, it could take a tad longer... or not. Physics can be a real joker sometimes!
To find the time taken by the ball to reach the ground, we need to determine when the height (H) of the ball is equal to zero since that indicates it has reached the ground.
Given the equation for the height of the ball, H = 29.4 + 24.5t - 4.9t², we set H to zero and solve for t:
0 = 29.4 + 24.5t - 4.9t²
Rearranging the equation, we get:
4.9t² - 24.5t - 29.4 = 0
Now we can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b² - 4ac))/(2a)
In the given equation, a = 4.9, b = -24.5, and c = -29.4. Substituting these values into the quadratic formula, we get:
t = (-(-24.5) ± √((-24.5)² - 4 * 4.9 * (-29.4))) / (2 * 4.9)
Simplifying further:
t = (24.5 ± √(600.25 + 576.72)) / 9.8
t = (24.5 ± √(1176.97)) / 9.8
Now, we can calculate the two possible values for t:
t₁ = (24.5 + √(1176.97)) / 9.8
t₂ = (24.5 - √(1176.97)) / 9.8
Using a calculator, we find that:
t₁ ≈ 4.8 seconds
t₂ ≈ -5.8 seconds
Since time cannot be negative in this scenario, we discard the negative value. Therefore, the time taken by the ball to reach the ground is approximately 4.8 seconds.
To find the time taken by the ball to reach the ground, we need to find the value of t when H equals 0. This is because when the ball hits the ground, the height H will be zero.
Given the equation H = 29.4 + 24.5t - 4.9t², we can set H to 0 and solve for t:
0 = 29.4 + 24.5t - 4.9t²
Rearranging the equation:
4.9t² - 24.5t - 29.4 = 0
Now, we can solve this quadratic equation to find the value of t. We can either factorize the quadratic equation or use the quadratic formula.
Let's use the quadratic formula: t = (-b ± √(b² - 4ac)) / (2a)
Here, a = 4.9, b = -24.5, and c = -29.4.
t = (-(-24.5) ± √((-24.5)² - 4 * 4.9 * -29.4)) / (2 * 4.9)
Simplifying:
t = (24.5 ± √(600.25 + 574.08)) / 9.8
t = (24.5 ± √(1174.33)) / 9.8
t = (24.5 ± 34.25) / 9.8
Now, we have two possible values for t:
t₁ = (24.5 + 34.25) / 9.8 ≈ 6.79 seconds
t₂ = (24.5 - 34.25) / 9.8 ≈ -1.02 seconds
Since time cannot be negative in this context, the time taken by the ball to reach the ground is approximately 6.79 seconds.