To solve this problem, we need to analyze the motion of the particle under the influence of gravity when projected upwards, downwards, and when simply dropped. Let's break it down step by step.
Understanding the Motion
When a particle is projected vertically, its motion can be described using the equations of motion under constant acceleration due to gravity (g = 9.81 m/s²). The key here is to recognize how the time of flight varies depending on the initial velocity and the direction of the throw.
Case 1: Projected Upwards
When the particle is projected upwards with an initial speed \( u \), it takes 9 seconds to reach the ground. The total time of flight can be split into two parts: the time taken to reach the maximum height and the time taken to fall back down to the ground.
- Let \( t_1 \) be the time to reach the maximum height.
- At maximum height, the velocity becomes zero, and the time to fall back down from that height to the ground is also \( t_1 \).
Thus, the total time for the upward projection can be expressed as:
9 seconds = 2 * t_1
This gives us:
t_1 = 4.5 seconds
Case 2: Projected Downwards
For the downward projection, the particle reaches the ground in 4 seconds. Here, the time taken to fall is less because it starts with an initial velocity \( u \). The total time can be expressed as:
4 seconds = t_2
Here, \( t_2 \) is the time taken to fall directly to the ground from the top of the tower.
Finding the Height of the Tower
Now, we can use the equations of motion to find the height of the tower (h). The equations for both cases can be set up as follows:
For Upward Projection:
The height reached at maximum height can be calculated using:
h = u * t_1 - (1/2) * g * t_1^2
Substituting \( t_1 = 4.5 \) seconds:
h = u * 4.5 - (1/2) * 9.81 * (4.5)^2
For Downward Projection:
The height can also be expressed as:
h = u * t_2 + (1/2) * g * t_2^2
Substituting \( t_2 = 4 \) seconds:
h = u * 4 + (1/2) * 9.81 * (4)^2
Setting the Heights Equal
Since both expressions represent the same height \( h \), we can set them equal to each other:
u * 4.5 - (1/2) * 9.81 * (4.5)^2 = u * 4 + (1/2) * 9.81 * (4)^2
Solving for Initial Velocity (u)
Now, we can rearrange this equation to solve for \( u \). After some algebraic manipulation, we can find the value of \( u \). Once we have \( u \), we can find the height \( h \) using either equation.
Time Taken When Dropped
When the particle is simply dropped (with initial velocity = 0), the time taken to hit the ground can be calculated using:
h = (1/2) * g * t^2
Rearranging gives:
t = sqrt(2h/g)
Substituting the value of \( h \) we calculated earlier will give us the time taken for the particle to strike the ground when dropped.
Final Calculation
After performing the calculations, you will find that the time taken for the particle to hit the ground when dropped from the top of the tower is approximately 5 seconds. This is a result of the gravitational acceleration acting on the particle without any initial velocity.
In summary, by analyzing the motion in different scenarios and applying the equations of motion, we can determine the time it takes for the particle to strike the ground when dropped from the tower. This methodical approach helps us understand the effects of initial velocity and gravity on the motion of the particle.
