Imagine a ball that is pushed off a desk completely horizontal and falls 9.47 m down to the floor. If it landed 3.32 m away from the edge of the desk, how fast was it moving in the horizonal direction just before it hit the ground? Round to 1 decimal.
The Correct Answer and Explanation is :
To determine the ball’s horizontal velocity just before it hit the ground, we need to break the problem into two parts: the vertical motion and the horizontal motion.
Step 1: Vertical Motion (Time to Fall)
We are given that the ball falls 9.47 meters vertically. The vertical motion is influenced only by gravity, so we use the kinematic equation for vertical displacement:
[
y = \frac{1}{2} g t^2
]
Where:
- (y = 9.47 \, \text{m}) (the vertical distance the ball falls),
- (g = 9.8 \, \text{m/s}^2) (the acceleration due to gravity),
- (t) is the time it takes for the ball to fall.
Rearranging the equation to solve for (t):
[
t = \sqrt{\frac{2y}{g}} = \sqrt{\frac{2 \times 9.47}{9.8}} \approx \sqrt{\frac{18.94}{9.8}} \approx \sqrt{1.93} \approx 1.39 \, \text{seconds}
]
So, the ball takes approximately 1.39 seconds to fall.
Step 2: Horizontal Motion (Horizontal Velocity)
The horizontal motion is independent of the vertical motion (ignoring air resistance), so the ball’s horizontal velocity remains constant. We know the ball lands 3.32 meters away from the edge of the desk. We can use the formula for distance traveled in horizontal motion:
[
\text{distance} = \text{velocity} \times \text{time}
]
Where:
- The distance traveled horizontally is 3.32 meters,
- The time taken to fall is 1.39 seconds (calculated earlier),
- The horizontal velocity ((v_x)) is what we need to find.
Rearranging the formula to solve for (v_x):
[
v_x = \frac{\text{distance}}{t} = \frac{3.32}{1.39} \approx 2.39 \, \text{m/s}
]
Final Answer:
The horizontal velocity of the ball just before it hit the ground is approximately 2.4 m/s.
Explanation:
The ball’s horizontal velocity is determined by how far it travels horizontally and how long it takes to fall. Since the time to fall is dictated by the vertical motion and the distance traveled horizontally is constant (assuming no air resistance), we can calculate the horizontal velocity using basic kinematic equations. The ball maintains the same horizontal speed throughout its fall, meaning its velocity just before impact is the same as when it left the desk, which we find to be approximately 2.4 m/s.