The braking distance of a car is directly proportional to te square of it's speed.
When the speed is p metres per second, the braking distance is 6m. When the speed is increased by 300%, find
(a) an expression for speed of the car
(b) the braking distance
(c) the % increase in the braking distance
Is there any fixed methods for this type of question? Thanks a lot!!! :)))))))
How did you find new speed?
Yes, there is a general approach to solving this type of problem. Here are the steps you can follow:
Step 1: Identify the variables and their relationship.
In this case, the variables are the speed of the car (p) and the braking distance (d). We are given that the braking distance (d) is directly proportional to the square of the speed (p^2).
Step 2: Write down the given information and the unknowns.
Given: When speed (p) is p meters per second, the braking distance (d) is 6m.
Unknowns: We need to find the expression for the speed of the car (a), the new braking distance (b), and the percentage increase in braking distance (c).
Step 3: Write down the proportional relationship.
Since braking distance (d) is directly proportional to the square of the speed (p^2), we can write this as:
d = kp^2, where k is the constant of proportionality.
Step 4: Use the given information to find the constant of proportionality (k).
Using the information given, we know that when p = p meters per second, d = 6 meters. Substituting these values into the proportional relationship equation, we get:
6 = k(p^2)
Step 5: Solve for the constant of proportionality (k).
To solve for k, we need to rearrange the equation and solve for k. First, divide both sides of the equation by p^2:
6/p^2 = k
Step 6: Substitute k into the proportional relationship equation.
Now that we know the value of k, we can substitute it back into the proportional relationship equation:
d = (6/p^2) * p^2
Simplifying the equation, we get:
d = 6
Step 7: Use the new information to solve for the unknowns.
(a) To find an expression for the new speed of the car, we are given that the speed is increased by 300%. This means the new speed is the original speed plus the 300% increase:
New speed = p + 300% of p
New speed = p + (3 * p)
New speed = 4p
So the expression for the speed of the car is 4p meters per second.
(b) To find the new braking distance, we substitute the new speed (4p) into the equation:
New braking distance = 6 * (4p)^2
Simplifying this equation, we get:
New braking distance = 6 * 16p^2
New braking distance = 96p^2 meters
(c) To find the percentage increase in the braking distance, we need to calculate the difference between the new braking distance and the original braking distance, and then calculate the percentage increase:
Percentage increase = ((New braking distance - Original braking distance) / Original braking distance) * 100%
Substituting the values, we get:
Percentage increase = ((96p^2 - 6) / 6) * 100%
Yes, there are fixed methods for solving this type of question. To find the expression for the speed of the car, braking distance, and the percentage increase in the braking distance, we can follow these steps:
Step 1: Establish the relationship between the speed and the braking distance.
Here, it is given that the braking distance of a car is directly proportional to the square of its speed. This can be written as:
Braking distance ∝ speed^2
Step 2: Use the given information to find the constant of proportionality.
In this case, we are given that when the speed is p meters per second, the braking distance is 6 meters. This can be written as:
6 ∝ p^2
To find the constant of proportionality, we divide both sides by p^2:
6/p^2 = k , where k is the constant of proportionality
Step 3: Find the expression for the speed of the car.
Now, we need to find the expression for the speed of the car when it is increased by 300%. We can represent this increased speed as 4p based on the given information.
So, the expression for the speed of the car can be written as:
Speed = 4p
Step 4: Find the braking distance when the speed is increased by 300%.
To find the braking distance, we substitute the expression for speed into the relationship we established in Step 1:
Braking distance ∝ (speed)^2
Since the speed is 4p, we have:
Braking distance ∝ (4p)^2
Braking distance ∝ 16p^2
Step 5: Calculate the braking distance.
Substitute the constant of proportionality (k) from Step 2 into the equation from Step 4:
16p^2 = 6/p^2
To solve for p, we can simplify the equation and rearrange it:
16p^4 = 6
p^4 = 6/16
p^4 = 3/8
p = (3/8)^(1/4) (taking the fourth root of both sides)
Now, we have the value of p, which represents the initial speed of the car.
Step 6: Calculate the braking distance when the speed is increased by 300%.
Using the expression for the speed of the car (4p), we can find the new braking distance:
Braking distance = 16p^2
Substituting the value of p from Step 5 into the equation, we get:
Braking distance = 16 * (3/8)^(1/2)
Step 7: Calculate the percentage increase in the braking distance.
To find the percentage increase, we compare the new braking distance with the initial braking distance and calculate the difference as a percentage.
Percentage increase = ((New braking distance - Initial braking distance) / Initial braking distance) * 100
Substitute the values into the equation to calculate the percentage increase in the braking distance.
This method can be applied to solve similar questions where there is a direct proportionality between two quantities and specific information is given to establish the relationship.
d = k v^2 so v = sqrt(d/k)
6 = k p^2 so p = sqrt (6/k)
new speed = 4 p = 4 sqrt (6/k)
new d = k v^2 = k*16(6/k) = 96 meters
96/6 = 16 times = 1600 %
or
new speed = 1600 percent of old speed
so
percent increase = 1600-100 = 1500% increase