A tower is 24m high. A stone is projected upwards from the tower with initial velocity of 24,5 m/s. Calculate how long it would take for the stone to reach the ground at the foot of the tower.
you know that the height at time t seconds is
h(t) = 24 + 24.5t - 4.9t^2
so, just solve for t when the stone hits the ground (its height is zero).
3,663sec & 1,337sec
To calculate the time it takes for the stone to reach the ground, we can use the equations of motion to find the time when the stone's height is zero.
The equation that relates height, initial velocity, time, and acceleration is:
h = ut + (1/2)at^2
where:
- h is the height,
- u is the initial velocity,
- t is the time taken, and
- a is the acceleration due to gravity (approximately -9.8 m/s^2).
In this case, the initial height of the stone is 24m, the initial velocity is 24.5 m/s, and the acceleration due to gravity is -9.8 m/s^2.
Since we are interested in the time it takes for the stone to reach the ground, which would be when the height is zero, we can rearrange the equation:
0 = 24 + 24.5t - (1/2)(9.8)t^2
Simplifying this equation, we have:
4.9t^2 - 24.5t - 24 = 0
Now we can solve this quadratic equation for t by factoring or using the quadratic formula.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4.9, b = -24.5, and c = -24.
Substituting these values into the formula, we have:
t = (-(-24.5) ± √((-24.5)^2 - 4 * 4.9 * (-24))) / (2 * 4.9)
Simplifying further:
t = (24.5 ± √(600.25 + 470.4)) / 9.8
t = (24.5 ± √(1070.65)) / 9.8
Calculating the square root of 1070.65, we get:
t = (24.5 ± 32.72) / 9.8
Now we have two solutions, one with the plus sign and one with the minus sign:
t₁ = (24.5 + 32.72) / 9.8
t₁ = 57.22 / 9.8
t₁ ≈ 5.84 seconds
t₂ = (24.5 - 32.72) / 9.8
t₂ = -8.22 / 9.8
t₂ ≈ -0.84 seconds
Since time cannot be negative in this context, we ignore the negative solution.
Therefore, it would take approximately 5.84 seconds for the stone to reach the ground at the foot of the tower.