the resistance to a motion of a car is partly constant and partly proportional to the square of the speed u. when the speed is 30km/hr the resistance is 190N and when the speed is 50km/hr the resistance is 350N. find what speed the resistance is 302.5N
Let's denote the constant resistance as R and the resistance that is proportional to the square of the speed as k·u², where k is a constant.
Given that when the speed is 30 km/hr, the resistance is 190 N, we can write the equation:
R + k(30 km/hr)² = 190 N
Similarly, when the speed is 50 km/hr, the resistance is 350 N, which gives us the equation:
R + k(50 km/hr)² = 350 N
Subtracting the first equation from the second equation, we get:
k(50 km/hr)² - k(30 km/hr)² = 350 N - 190 N
Simplifying,
k(50² - 30²) = 160
k(2500 - 900) = 160
k(1600) = 160
k = 160 / 1600
k = 0.1
Now we can substitute k into one of the earlier equations to find R:
R + 0.1(30 km/hr)² = 190 N
R + 0.1(900 km²/hr²) = 190 N
R + 90 N = 190 N
R = 190 N - 90 N
R = 100 N
Finally, we can find the speed at which the resistance is 302.5 N:
R + k·u² = 302.5 N
100 N + 0.1u² = 302.5 N
0.1u² = 302.5 N - 100 N
0.1u² = 202.5 N
u² = 202.5 N / 0.1
u² = 2025
Taking the square root of both sides:
u = √(2025)
u ≈ 45 km/hr
Therefore, the speed at which the resistance is 302.5 N is approximately 45 km/hr.
To solve this problem, we need to break down the resistance into its constant and proportional components. Let's denote the constant resistance as R0 and the proportional resistance as R1.
Given information:
- Speed u1 = 30 km/hr
- Resistance R1 = 190 N
From the given information, we can set up the following equation:
R1 = R0 + k * u1^2, where k is the proportional constant.
Substituting the values:
190 = R0 + k * (30)^2
Now let's use the second set of given information:
- Speed u2 = 50 km/hr
- Resistance R2 = 350 N
We can set up a similar equation:
R2 = R0 + k * u2^2
Substituting the values:
350 = R0 + k * (50)^2
Now we have two equations with two unknowns (R0 and k):
190 = R0 + k * 900,
350 = R0 + k * 2500
To solve these equations, we can subtract the first equation from the second equation:
350 - 190 = (R0 + k * 2500) - (R0 + k * 900)
160 = k * 2500 - k * 900
Simplifying further:
160 = 1600k
Dividing both sides by 1600:
k = 0.1
Now that we have the value of k, we can substitute it back into either of the original equations to find R0. Let's use the first equation:
190 = R0 + 0.1 * (30)^2
190 = R0 + 0.1 * 900
190 = R0 + 90
R0 = 100
So the constant resistance (R0) is 100 N.
Now let's find the resistance (R) when the resistance is 302.5 N. Let's denote this speed as u:
302.5 = R0 + k * u^2
302.5 = 100 + 0.1 * u^2
Subtracting 100 from both sides:
202.5 = 0.1 * u^2
Dividing both sides by 0.1:
u^2 = 2025
Taking the square root of both sides:
u = √2025
u = 45 km/hr
Therefore, when the resistance is 302.5 N, the speed is approximately 45 km/hr.
To find the speed at which the resistance is 302.5N, we need to solve the problem by setting up a system of equations.
Let's assume the constant resistance is represented by "a" and the resistance proportional to the square of the speed is represented by "b."
Given:
When the speed is 30 km/hr, the resistance is 190N.
When the speed is 50 km/hr, the resistance is 350N.
Using the given data, we can set up the following equations:
a + b(30^2) = 190 ...(Equation 1)
a + b(50^2) = 350 ...(Equation 2)
Now, we can solve these two equations simultaneously.
First, expand the squared terms:
a + b(900) = 190 ...(Equation 1')
a + b(2500) = 350 ...(Equation 2')
Next, re-arrange the equations in terms of "a":
a = 190 - 900b ...(Equation 3)
a = 350 - 2500b ...(Equation 4)
Setting Equation 3 and Equation 4 equal to each other:
190 - 900b = 350 - 2500b
Now, solve for "b":
900b - 2500b = 350 - 190
-1600b = 160
b = -160/(-1600)
b = 0.1
Substitute the value of "b" into Equation 3 or Equation 4 to find the value of "a"(constant resistance):
a = 190 - 900(0.1)
a = 190 - 90
a = 100
So, the constant resistance (a) is 100 N, and the resistance proportional to the square of the speed (b) is 0.1.
Now, we can use these values in Equation 1 to find the speed at which the resistance is 302.5 N:
a + b(u^2) = 302.5
Plugging in the values we found:
100 + 0.1(u^2) = 302.5
Rearranging the equation:
0.1(u^2) = 202.5
Dividing both sides by 0.1:
u^2 = 2025
Taking the square root of both sides:
u = √2025
Simplifying:
u = 45
Therefore, the speed at which the resistance is 302.5 N is 45 km/hr.