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.