A ball is thrown horizontally from a building of height 80m. the kine joining the top of the building to the point where it hits the ground makes an angle of 45 degrees with the ground. what is the initial velocity of the ball?
To find the initial velocity of the ball, we need to make use of the relationship between the horizontal distance traveled by the ball and the time it takes to reach the ground.
Let's break down the given information:
- The building height is 80m.
- The angle between the ground and the line joining the top of the building to the point where the ball hits the ground is 45 degrees.
Based on this information, we can divide the motion of the ball into horizontal and vertical components.
Horizontal motion:
The ball is thrown horizontally, indicating that there is no acceleration in the horizontal direction. Therefore, the horizontal component of the initial velocity remains constant throughout the motion.
Vertical motion:
The ball is subject to vertical acceleration due to gravity. The motion can be divided into two parts: upward and downward. The time taken for the upward motion is the same as the time taken for the downward motion.
Now, let's calculate the time taken for the ball to reach the ground using the vertical motion component.
Using the equation of motion for vertical motion:
d = ut + (1/2)at^2
Where:
d = vertical displacement (80m)
u = initial vertical velocity (0 since the ball is thrown horizontally)
a = acceleration due to gravity (-9.8 m/s^2)
t = time
Substituting the given values into the equation, we get:
80 = 0t + (1/2)(-9.8)t^2
80 = -4.9t^2
Simplifying the equation, we have:
4.9t^2 = -80
Dividing both sides by 4.9 to solve for t^2, we get:
t^2 = -80 / 4.9
Taking the square root of both sides to solve for t, we get:
t = √(-80 / 4.9)
We need to take the positive value of t, as time cannot be negative.
Now that we have the time taken for the ball to reach the ground, we can find the horizontal distance traveled by the ball using the equation:
horizontal distance = horizontal velocity * time
Since the ball was thrown horizontally, the horizontal velocity is the same as the initial velocity. Therefore, the horizontal distance traveled is the same as the initial velocity.
Given that the angle between the line joining the top of the building to the point where the ball hits the ground and the ground is 45 degrees, we can use trigonometry to find the horizontal velocity.
Using the formula:
horizontal velocity = initial velocity * cos(angle)
Substituting the given angle (45 degrees), we get:
horizontal velocity = initial velocity * cos(45)
Now, we can equate the horizontal distance and the horizontal velocity multiplied by the time calculated earlier:
initial velocity = horizontal distance / (horizontal velocity * time)
Substituting the values we have determined, we can calculate the initial velocity of the ball.