Chapter 8: Problem 67

A body dropped from a height of $5 \mathrm{~m}$, reaches the ground in $1 \mathrm{~s}$. If it takes $2 \mathrm{~s}$ to reach the ground, find the height from which it is dropped.

Short Answer

Expert verified

Answer: The body must be dropped from a height of 20 meters to reach the ground in 2 seconds.

Step by step solution

Write down the given values

We are given: - The initial height $h_1 = 5 \mathrm{~m}$ - The time taken to reach the ground for the first case $t_1 = 1 \mathrm{~s}$ - The time taken to reach the ground for the second case $t_2 = 2 \mathrm{~s}$

Use the displacement formula for initial case

For the initial case, we use the formula of displacement for a falling object under uniform acceleration. When a body is dropped, the initial velocity $u = 0$. The displacement formula is: $$ s = ut + \frac{1}{2}at^2 $$ Where - $s$ is the displacement - $u$ is the initial velocity - $t$ is time taken to reach the ground - $a$ is the acceleration due to gravity (approx $9.81\,\text{m/s}^2$, but in most high school problems, $a=10\,\text{m/s}^2$ is used) Using the given values for the initial case, we get: $$ 5 = 0 \cdot 1 + \frac{1}{2}a \cdot 1^2 $$

Calculate the acceleration due to gravity

By solving the equation from step 2, we can calculate the acceleration due to gravity $a$: $$ a = 2 \cdot 5 = 10 \,\text{m/s}^2 $$

Use the displacement formula for the second case

Now we have the acceleration due to gravity, we can use the displacement formula from step 2 to find the height from which the body is dropped in the second case. Let this new height be $h_2$: $$ h_2 = 0 \cdot 2 + \frac{1}{2} \cdot 10 \cdot 2^2 $$

Calculate the new height

Now, we calculate the height $h_2$ from the equation obtained in step 4: $$ h_2 = \frac{1}{2} \cdot 10 \cdot 4$$ $$ h_2 = 20 \,\text{m} $$ The body is dropped from a height of $20 \,\text{m}$ to reach the ground in $2 \,\text{s}$.

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A body dropped from a height of \(5 \mathrm{~m}\), reaches the ground in \(1 \mathrm{~s}\). If it takes \(2 \mathrm{~s}\) to reach the ground, find the height from which it is dropped.

Short Answer

Step by step solution

Write down the given values

Use the displacement formula for initial case

Calculate the acceleration due to gravity

Use the displacement formula for the second case

Calculate the new height

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