Calculating the Time Taken for a Body Thrown Vertically Upward from a Tower to Reach the Ground
Calculating the Time Taken for a Body Thrown Vertically Upward from a Tower to Reach the Ground
A classic problem in physics involves understanding the motion of a body when it's thrown vertically upward. This problem examines the specific scenario where a body is thrown from the top of a tower and determines the time taken to reach the ground below. Let's explore this in detail.
Problem Statement
A body is thrown vertically upward from the top of a tower 40.0 m high with an initial velocity of 10.0 m/s. The acceleration due to gravity is 10.0 m/s2. The objective is to find the time taken for the body to reach the ground.
Understanding the Motion
The motion of the body can be divided into two parts:
The upward motion from the point of projection until it reaches the maximum height. The downward motion from the maximum height to the ground.Step-by-Step Solution
First, let's consider the upward motion part.
Upward Motion
At the maximum height, the velocity of the body becomes zero. Using the kinematic equation:
v u - gt
Setting v 0 and u 10.0 m/s, we can find the time taken for the body to reach its maximum height.
0 10.0 - 10t
10t 10.0
t 1.0 s
Using the same time to reach the maximum height, we can calculate the maximum height reached by the body.
H ut - 1/2 gt2
H 10.0(1.0) - 1/2(10.0)(1.0)2
H 10.0 – 5.0 5.0 m
So, the maximum height above the top of the tower is 5.0 meters. Therefore, the total height from the ground to the maximum point is:
H_total 40.0 5.0 45.0 m
Next, we need to find the time taken for the body to fall from its maximum height to the ground.
Downward Motion
Using the kinematic equation:
s ut 1/2 at2
where s 45.0 m, u 0 m/s (since the velocity at the maximum height is zero, and it now falls freely under gravity), and a 10.0 m/s2.
45.0 0t 1/2(10.0)t2
45.0 5.0t2
t2 45.0 / 5.0 9.0
t 3.0 s
Adding the time taken for the upward and downward motions, we get the total time taken for the body to reach the ground:
Total time 1.0 s 3.0 s 4.0 s
Verification Using the Quadratic Equation
To confirm, let's use the vertical displacement equation:
yt -9.8/2t^2 10t 40
Setting yt 0 to find the time the body reaches the ground:
0 -4.9t^2 10t 40
Solving the quadratic equation, we get:
t -2.01 or t 4.05
Only the positive solution makes sense in this context, so:
t 4.05 s
Conclusion
The time taken for the body to reach the ground is 4.05 seconds. This confirms that the body indeed hits the ground after a total of 4.05 seconds.