Exercise 16
(Unless stated otherwise, take the acceleration due to gravity as g=10ms)

1. A particle is projected horizontally with a velocity of 40 ms from the top of a tower 80.5m above the level ground. Find how far from the bottom of the tower the particle is when it hits the ground.

160.48 ​

Explanation

Problem Analysis We are given a projectile motion problem where a particle is projected horizontally from a tower. We need to find the horizontal distance the particle travels before hitting the ground.

Finding the Time of Flight First, we need to find the time it takes for the particle to hit the ground. Since the initial vertical velocity is zero, we can use the following kinematic equation to find the time:


y = y 0 ​ + v 0 y ​ t + 2 1 ​ a t 2
where:

y is the final vertical position (0 m, ground level)
y 0 ​ is the initial vertical position (80.5 m)
v 0 y ​ is the initial vertical velocity (0 m/s)
a is the acceleration due to gravity (10 m/s 2 )
t is the time

Plugging in the values, we get:
0 = 80.5 + 0 ⋅ t + 2 1 ​ ( 10 ) t 2
0 = 80.5 + 5 t 2
Solving for t 2 :
5 t 2 = − 80.5
t 2 = 5 80.5 ​ = 16.1
t = 16.1 ​ ≈ 4.012 seconds

Calculating the Horizontal Distance Now that we have the time, we can find the horizontal distance the particle travels. Since there is no horizontal acceleration (we are neglecting air resistance), the horizontal velocity remains constant. We can use the following equation:

x = v x ​ ⋅ t
where:

x is the horizontal distance
v x ​ is the horizontal velocity (40 m/s)
t is the time (4.012 seconds)

Plugging in the values, we get:
x = 40 ⋅ 4.012 ≈ 160.48 meters

Final Answer Therefore, the particle hits the ground approximately 160.48 meters from the bottom of the tower.

Examples
Understanding projectile motion is crucial in various real-world scenarios, such as in sports like baseball or basketball, where players need to calculate the trajectory of a ball to make accurate throws or shots. Similarly, engineers use these principles to design systems that involve launching objects, like in delivery systems or even in understanding the trajectory of water in irrigation systems. By understanding the initial conditions and applying the equations of motion, one can predict the landing point of a projectile, optimizing performance and safety.

Answered by GinnyAnswer | 2025-07-03

The particle hits the ground approximately 160.48 meters from the bottom of the tower after being projected horizontally from a height of 80.5 meters. To find this, the time of flight was first calculated, followed by the horizontal distance traveled. Both calculations utilized the principles of projectile motion.
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Answered by Anonymous | 2025-07-04