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Answer :
Sure, let’s solve the problem step by step.
### Step 1: Understand the problem
A ball is rolled off a horizontal table with a speed of [tex]\( 1.6 \, \text{m/s} \)[/tex] and the table is [tex]\( 1.2 \, \text{m} \)[/tex] high. We need to calculate how far the ball travels horizontally before it hits the ground.
### Step 2: Find the time it takes for the ball to fall to the ground
The time it takes for the ball to fall is determined by the vertical motion. The motion is under the influence of gravity, so we use the following kinematic equation:
[tex]\[ d = \frac{1}{2} g t^2 \][/tex]
where:
- [tex]\( d \)[/tex] is the vertical distance (height of the table) [tex]\( = 1.2 \, \text{m} \)[/tex]
- [tex]\( g \)[/tex] is the acceleration due to gravity [tex]\( = 9.8 \, \text{m/s}^2 \)[/tex]
- [tex]\( t \)[/tex] is the time in seconds
Rearrange the equation to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \sqrt{\frac{2d}{g}} \][/tex]
Substitute the values:
[tex]\[ t = \sqrt{\frac{2 \times 1.2}{9.8}} \][/tex]
After calculating this expression, we find that:
[tex]\[ t \approx 0.4949 \, \text{seconds} \][/tex]
### Step 3: Calculate the horizontal distance
To find how far the ball travels horizontally, we use the horizontal speed of the ball and the time calculated above. The horizontal distance [tex]\( d_{\text{horizontal}} \)[/tex] can be found using the formula:
[tex]\[ d_{\text{horizontal}} = \text{speed}_{\text{horizontal}} \times t \][/tex]
Given:
- The horizontal speed [tex]\( = 1.6 \, \text{m/s} \)[/tex]
- The time [tex]\( t \approx 0.4949 \, \text{seconds} \)[/tex]
So,
[tex]\[ d_{\text{horizontal}} = 1.6 \times 0.4949 \][/tex]
After calculating this multiplication, we find the horizontal distance:
[tex]\[ d_{\text{horizontal}} \approx 0.7918 \, \text{meters} \][/tex]
### Conclusion
The ball travels approximately [tex]\( 0.7918 \, \text{meters} \)[/tex] in the horizontal direction before it hits the floor.
### Step 1: Understand the problem
A ball is rolled off a horizontal table with a speed of [tex]\( 1.6 \, \text{m/s} \)[/tex] and the table is [tex]\( 1.2 \, \text{m} \)[/tex] high. We need to calculate how far the ball travels horizontally before it hits the ground.
### Step 2: Find the time it takes for the ball to fall to the ground
The time it takes for the ball to fall is determined by the vertical motion. The motion is under the influence of gravity, so we use the following kinematic equation:
[tex]\[ d = \frac{1}{2} g t^2 \][/tex]
where:
- [tex]\( d \)[/tex] is the vertical distance (height of the table) [tex]\( = 1.2 \, \text{m} \)[/tex]
- [tex]\( g \)[/tex] is the acceleration due to gravity [tex]\( = 9.8 \, \text{m/s}^2 \)[/tex]
- [tex]\( t \)[/tex] is the time in seconds
Rearrange the equation to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \sqrt{\frac{2d}{g}} \][/tex]
Substitute the values:
[tex]\[ t = \sqrt{\frac{2 \times 1.2}{9.8}} \][/tex]
After calculating this expression, we find that:
[tex]\[ t \approx 0.4949 \, \text{seconds} \][/tex]
### Step 3: Calculate the horizontal distance
To find how far the ball travels horizontally, we use the horizontal speed of the ball and the time calculated above. The horizontal distance [tex]\( d_{\text{horizontal}} \)[/tex] can be found using the formula:
[tex]\[ d_{\text{horizontal}} = \text{speed}_{\text{horizontal}} \times t \][/tex]
Given:
- The horizontal speed [tex]\( = 1.6 \, \text{m/s} \)[/tex]
- The time [tex]\( t \approx 0.4949 \, \text{seconds} \)[/tex]
So,
[tex]\[ d_{\text{horizontal}} = 1.6 \times 0.4949 \][/tex]
After calculating this multiplication, we find the horizontal distance:
[tex]\[ d_{\text{horizontal}} \approx 0.7918 \, \text{meters} \][/tex]
### Conclusion
The ball travels approximately [tex]\( 0.7918 \, \text{meters} \)[/tex] in the horizontal direction before it hits the floor.
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