When a ball is allowed to fall, the time it takes to fall any distance varies as the square root of the distance and it takes 4 seconds to fall 78.40 m. How long would it take to fall 122.50 m?
5 seconds
Understanding the Relationship Between Fall Time and Distance
The question describes a scenario where the time it takes for a ball to fall is related to the distance it falls. We are told that the time varies as the square root of the distance. This is a common relationship encountered in physics, particularly in problems involving constant acceleration like free fall under gravity (assuming negligible air resistance).
Mathematically, this relationship can be written as:
\( t \propto \sqrt{d} \)
Where:
- \( t \) represents the time taken to fall.
- \( d \) represents the distance fallen.
- \( \propto \) means "is proportional to".
To turn this proportionality into an equation, we introduce a constant of proportionality, let's call it \( k \):
\( t = k\sqrt{d} \)
We are given one specific case: it takes 4 seconds to fall 78.40 m. We can use this information to find the value of the constant \( k \).
\( 4 \text{ s} = k \times \sqrt{78.40 \text{ m}} \)
We can also use the fact that the ratio \( \frac{t}{\sqrt{d}} \) is constant for any distance fallen. So, if \( t_1 \) is the time to fall distance \( d_1 \), and \( t_2 \) is the time to fall distance \( d_2 \), we have:
\( \frac{t_1}{\sqrt{d_1}} = \frac{t_2}{\sqrt{d_2}} \)
This ratio method is often simpler as it avoids calculating the constant \( k \) explicitly.
Applying the Ratio Method to Solve for Fall Time
We are given the following values:
- Time \( t_1 = 4 \) seconds
- Distance \( d_1 = 78.40 \) meters
- New distance \( d_2 = 122.50 \) meters
We need to find the new time \( t_2 \) to fall the distance \( d_2 \).
Using the ratio equation:
\( \frac{t_1}{\sqrt{d_1}} = \frac{t_2}{\sqrt{d_2}} \)
Substitute the known values:
\( \frac{4 \text{ s}}{\sqrt{78.40 \text{ m}}} = \frac{t_2}{\sqrt{122.50 \text{ m}}} \)
Now, we need to solve for \( t_2 \). We can rearrange the equation:
\( t_2 = 4 \text{ s} \times \frac{\sqrt{122.50 \text{ m}}}{\sqrt{78.40 \text{ m}}} \)
We can simplify the square roots by combining them:
\( t_2 = 4 \text{ s} \times \sqrt{\frac{122.50 \text{ m}}{78.40 \text{ m}}} \)
Step-by-Step Calculation
Let's perform the calculation step-by-step:
- Calculate the ratio of the distances inside the square root:
\( \frac{122.50}{78.40} \)
\( 122.50 \div 78.40 = 1.5625 \)
- Take the square root of this ratio:
\( \sqrt{1.5625} \)
\( \sqrt{1.5625} = 1.25 \)
- Multiply this result by the initial time \( t_1 \):
\( t_2 = 4 \text{ s} \times 1.25 \)
\( t_2 = 5 \text{ s} \)
So, it would take 5 seconds for the ball to fall 122.50 m.
| Quantity | Symbol | Value 1 | Value 2 |
|---|---|---|---|
| Time | \( t \) | \( t_1 = 4 \) s | \( t_2 = ? \) |
| Distance | \( d \) | \( d_1 = 78.40 \) m | \( d_2 = 122.50 \) m |
Revision Table: Key Concepts for Fall Time Calculation
| Concept | Description | Formula/Relation |
|---|---|---|
| Proportionality | Two quantities vary together such that their ratio remains constant. | \( y \propto x \) or \( y = kx \) |
| Square Root Variation | A quantity varies directly as the square root of another quantity. | \( t \propto \sqrt{d} \) or \( t = k\sqrt{d} \) |
| Ratio Method | For proportional relationships, the ratio of corresponding quantities is constant. | \( \frac{y_1}{x_1} = \frac{y_2}{x_2} \) or \( \frac{t_1}{\sqrt{d_1}} = \frac{t_2}{\sqrt{d_2}} \) |
Additional Information on Free Fall and Time
The relationship \( t \propto \sqrt{d} \) is specifically true for objects falling under constant acceleration from rest, like in free fall near the Earth's surface (ignoring air resistance). The actual formula relating distance and time in free fall from rest is \( d = \frac{1}{2}gt^2 \), where \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \)).
From \( d = \frac{1}{2}gt^2 \), we can rearrange to solve for \( t \):
\( t^2 = \frac{2d}{g} \)
\( t = \sqrt{\frac{2d}{g}} \)
\( t = \sqrt{\frac{2}{g}} \times \sqrt{d} \)
Comparing this to \( t = k\sqrt{d} \), we see that the constant of proportionality \( k \) is equal to \( \sqrt{\frac{2}{g}} \). Using \( g = 9.8 \text{ m/s}^2 \):
\( k = \sqrt{\frac{2}{9.8}} = \sqrt{\frac{1}{4.9}} \approx \sqrt{0.204} \approx 0.4517 \)
This matches the value of \( k \) we would have calculated using the first method \( k = \frac{4}{\sqrt{78.40}} \approx \frac{4}{8.854} \approx 0.4517 \). This confirms the stated relationship is consistent with the physics of free fall.
Similar Questions
If three positive numbers are in the ratio 2 : 3 : 5 and the sum of their squares is 1368, then what is sum of all the numbers ?
The annual incomes of two persons are in the ratio 9 ∶ 7 and their expenses are in the ratio 4 ∶ 3. If each of them saves Rs. 2000 per year, what is the difference in their annual incomes?
In a class of 49 students, the ratio of girls to boys is 4 ∶ 3. If 4 girls leave the class, the ratio of girls to boys would be
Incomes of Mahesh and Kamal are in the ratio 1 : 2 and their expenses are in the ratio 1 : 3. Which one of the following is correct?
In an office, one-third of the workers are women, half of the women are married, and one-third of the married women have children. If three-fourth of the men are married and one-third of the married men have children, then what is the ratio of married women to married men?
If (2ab - b2) : (6a2 - ab) = 1 : 6, then what is the value of (a + b) : (a - b)?
The incomes of A, B and C are in the ratio 7 ∶ 9 ∶ 12 and their expenditures are in the ratio 8 ∶ 9 ∶ 15. If A's saving is one-fourth of his income, then the ratio of savings of A, B and C is
- If (a + b + c): d = 5: 1, (b + c + d): a = 7: 1 and, (c + d + a): b = 3: 1 then find a: b: c: d?
Rs. 120 is distributed among A, B and C so that A’s share is Rs. 20 more than B’s and Rs. 20 less than C’s. What is the B’s share?
In the following table of inverse variations, what are the values of A, B and C respectively?
M
15
-6
2
C
N
-4
A
B
60
Important Questions from Simple Ratios
If x : y = 5 : 2, then the value of (8x + 9y) : (8x + 2y) is:
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
Two numbers are in the ratio of 9 : 7. If the larger number is 56 more than one-seventh of the smaller, then what is the sum of the two numbers?
The ratio of two numbers A and B is 5: 8. If 5 is added to each of A and B, then the ratio becomes 2 : 3. The difference between A and B is:
In population of a city the ratio of men and women is 12 : 11. If total population of that city is 4,60,000, then the population of women is:
