mobiles travel is often estimated from the length proportional to the skid marks. Assume that V is directly proportional to the square root of L, where V is the speed in miles per hour and L is the length of the skid marks. Find a formula that involves a constant of proportionality. Given that V = 42 mph when L = 625 feet, find the value of k in part (a). (c) Find the initial speed when the skid marks are 242 feet long. -13 points SwokPreCalc13 3.6.021. The period T of a simple pendulum is directly proportional to the square root of the length L of the pendulum. If the period is 3.5 seconds, find the value of k in part (a). Find the length of a pendulum that has a period of 5 seconds. -12 points SwokPreCalc13 3.6.028 The ideal gas law states that the volume V that a gas occupies is directly proportional to the product of the number of moles n, the gas constant R, and the temperature T (in Kelvin). Express V in terms of n, R, and T. (b) What is the effect on volume if the number of moles is doubled and both the temperature and pressure are reduced by half?
-13 points SwokPreCalc13 – 6.025 The speed at which Bxpre auton mobiles travel is often estimated from the length proportional to the skid marks. Assume that V is directly proportional to the square root of L, where V is the speed in miles per hour and L is the length of the skid marks. Find a formula that involves a constant of proportionality. Given that V = 42 mph when L = 625 feet, find the value of k in part (a). (c) Find the initial speed when the skid marks are 242 feet long. -13 points SwokPreCalc13 3.6.021. The period T of a simple pendulum is directly proportional to the square root of the length L of the pendulum. If the period is 3.5 seconds, find the value of k in part (a). Find the length of a pendulum that has a period of 5 seconds. -12 points SwokPreCalc13 3.6.028 The ideal gas law states that the volume V that a gas occupies is directly proportional to the product of the number of moles n, the gas constant R, and the temperature T (in Kelvin). Express V in terms of n, R, and T. (b) What is the effect on volume if the number of moles is doubled and both the temperature and pressure are reduced by half
The Correct Answer and Explanation is:
Problem 1: Skid Marks and Speed
Given:
Speed VV is directly proportional to the square root of the skid mark length LL.
So, the relationship is: V=kLV = k\sqrt{L}
Given:
V=42V = 42 mph when L=625L = 625 ft
Plug into the formula: 42=k625⇒42=k(25)⇒k=4225=1.6842 = k\sqrt{625} \Rightarrow 42 = k(25) \Rightarrow k = \frac{42}{25} = 1.68
Final formula: V=1.68LV = 1.68\sqrt{L}
(c) Find speed when L=242L = 242: V=1.68242≈1.68×15.556=26.13 mphV = 1.68\sqrt{242} \approx 1.68 \times 15.556 = 26.13 \text{ mph}
Problem 2: Pendulum Period
Given:
TT is directly proportional to L\sqrt{L}: T=kLT = k\sqrt{L}
Given:
T=3.5T = 3.5 seconds for some unknown LL
We’ll use this later.
Now solve for the length when T=5T = 5:
Use the same relationship: T1T2=L1L2⇒(3.55)2=L1L2⇒(710)2=L1L2⇒49100=L1L2⇒L2=10049L1\frac{T_1}{T_2} = \frac{\sqrt{L_1}}{\sqrt{L_2}} \Rightarrow \left(\frac{3.5}{5}\right)^2 = \frac{L_1}{L_2} \Rightarrow \left(\frac{7}{10}\right)^2 = \frac{L_1}{L_2} \Rightarrow \frac{49}{100} = \frac{L_1}{L_2} \Rightarrow L_2 = \frac{100}{49}L_1
So if you know L1L_1, you can find L2L_2, or vice versa.
Problem 3: Ideal Gas Law
(a) Formula:
The volume VV is directly proportional to nRTnRT: V=k⋅nRTV = k \cdot nRT
For ideal gases, the constant kk is 1P\frac{1}{P} (inverse of pressure), so: V=nRTPV = \frac{nRT}{P}
(b) Effect of changing variables:
- If nn doubles → volume doubles
- If TT is halved → volume is halved
- If PP is halved → volume doubles (since V∝1PV \propto \frac{1}{P})
Net effect: Vnew=2n⋅12T⋅112P=2n⋅12T⋅2=4nT⇒Volume doubles overallV_{\text{new}} = 2n \cdot \frac{1}{2}T \cdot \frac{1}{\frac{1}{2}P} = 2n \cdot \frac{1}{2}T \cdot 2 = 4nT \Rightarrow \text{Volume doubles overall}
Combined Explanation:
These problems focus on direct variation, where one quantity changes in proportion to another. In the first problem, the speed of a vehicle is proportional to the square root of the skid mark length. By using the known values, we determined the constant of proportionality k=1.68k = 1.68, leading to the equation V=1.68LV = 1.68\sqrt{L}. When the skid marks are 242 feet long, we estimate the speed at approximately 26.13 mph. This method is widely used in accident reconstruction.
In the pendulum problem, the period TT is directly proportional to the square root of the length LL. This reflects how gravity affects motion: longer pendulums swing more slowly. By understanding the ratio between periods and lengths, one can determine unknown values efficiently using proportional relationships.
The ideal gas law expresses volume VV as directly proportional to the product nRTnRT, and inversely to pressure PP. When the number of moles doubles, temperature halves, and pressure halves, the net result is a doubling of the volume. This is due to the interplay of direct and inverse variations, a foundational concept in thermodynamics.
Overall, these problems illustrate how proportionality constants help model real-world phenomena across physics, chemistry, and engineering.
