a ball is throw horigentally from a height of 90m with a speed of 4.5m/s then find the time when it is at the height of 45m from the ground

To determine the time it takes for a ball thrown horizontally from a height of 90 meters to reach a height of 45 meters, we can break the problem down into two parts: the vertical motion and the horizontal motion. Since the ball is thrown horizontally, its initial vertical velocity is zero, and we will focus on how long it takes to fall from 90 meters to 45 meters.

Understanding Vertical Motion

The vertical motion of the ball is influenced by gravity, which accelerates the ball downwards at approximately 9.81 m/s². We can use the following kinematic equation to find the time it takes to fall a certain distance:

• Equation: h = h₀ + v₀t + (1/2)gt²

In this equation:

• h is the final height (45 m)
• h₀ is the initial height (90 m)
• v₀ is the initial vertical velocity (0 m/s)
• g is the acceleration due to gravity (approximately 9.81 m/s²)
• t is the time in seconds

Setting Up the Equation

We can rearrange the equation to find the time:

• First, substitute the known values:
• 45 = 90 + 0*t - (1/2)(9.81)t²

This simplifies to:

• 45 = 90 - 4.905t²

Now, we can rearrange it to isolate the term with time:

• 4.905t² = 90 - 45
• 4.905t² = 45

Solving for Time

Next, we can solve for t²:

• t² = 45 / 4.905
• t² ≈ 9.16

Now, take the square root to find t:

• t ≈ √9.16
• t ≈ 3.02 seconds

Final Result

Thus, the time it takes for the ball to reach a height of 45 meters from the ground is approximately 3.02 seconds. This calculation shows how the vertical motion under the influence of gravity can be analyzed separately from the horizontal motion, which remains constant at 4.5 m/s in this case.