To find the total time taken by the object to reach the ground when projected vertically upwards, we can use the equations of motion.
First, let's determine the initial velocity (u) of the object as it is projected vertically upwards. Given that the initial velocity is 20 m/s and the object is moving vertically upwards, the initial velocity will be negative (-20 m/s).
The object is projected from a tower of height 60 m, and we want to find the total time taken to reach the ground. We can use the equation of motion:
s = ut + (1/2)at^2
Here, s is the displacement (change in height), u is the initial velocity, t is the time taken, and a is the acceleration due to gravity.
Since the object is moving vertically upwards, the acceleration due to gravity will be negative (-9.8 m/s^2). The displacement (s) is -60 m (negative because the object is moving upwards).
Substituting the known values into the equation:
-60 m = (-20 m/s) * t + (1/2) * (-9.8 m/s^2) * t^2
Simplifying the equation:
-60 m = -20 m/s * t - 4.9 m/s^2 * t^2
Rearranging the equation:
4.9 t^2 + 20 t - 60 = 0
Now, we can solve this quadratic equation to find the values of t. Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
where a = 4.9, b = 20, and c = -60.
Substituting the values into the formula:
t = (-20 ± √(20^2 - 4 * 4.9 * (-60))) / (2 * 4.9)
Simplifying the equation:
t = (-20 ± √(400 + 1176)) / 9.8
t = (-20 ± √(1576)) / 9.8
t ≈ (-20 ± 39.7) / 9.8
This gives us two possible solutions for time, t1 and t2:
t1 ≈ (-20 + 39.7) / 9.8 ≈ 1.97 s (ignoring the negative solution)
t2 ≈ (-20 - 39.7) / 9.8 ≈ -6.2 s
Since time cannot be negative in this context, we discard the negative solution.
Therefore, the total time taken by the object to reach the ground is approximately 1.97 seconds.