Formula for Distance Under Uniform Acceleration

Oct 13, 2025 by talynt.org 68 views

Understanding the Problem

To solve this problem, we need to understand the concept of uniform acceleration and the formula for calculating distance traveled under such conditions. The car starts with an initial speed of 40 km/h and increases its speed by 5 km/h every hour. This means that the car's speed is not constant, but rather increases linearly with time.

Formula for Distance Under Uniform Acceleration

The formula for calculating distance traveled under uniform acceleration is given by:

d = ut + (1/2)at^2

where:

d is the distance traveled
u is the initial speed
t is the time
a is the acceleration

However, in this case, the acceleration is not constant, but rather increases by 5 km/h every hour. To solve this problem, we need to use a different approach.

Calculating Distance Traveled

Let's break down the problem into smaller time intervals, each of 1 hour. In the first hour, the car travels at a speed of 40 km/h. In the second hour, the car travels at a speed of 45 km/h, and so on.

We can calculate the distance traveled in each hour as follows:

Hour 1: distance = speed × time = 40 km/h × 1 h = 40 km
Hour 2: distance = speed × time = 45 km/h × 1 h = 45 km
Hour 3: distance = speed × time = 50 km/h × 1 h = 50 km
...
Hour n: distance = speed × time = (40 + 5(n-1)) km/h × 1 h = (40 + 5(n-1)) km

We can see that the distance traveled in each hour increases by 5 km.

Finding the Total Distance Traveled

To find the total distance traveled, we need to sum up the distances traveled in each hour. We can do this by using the formula for the sum of an arithmetic series:

d = n/2 × (a + l)

where:

d is the total distance traveled
n is the number of hours
a is the initial distance (40 km)
l is the last distance (385 km)

However, we need to find the value of n, which is the number of hours it takes to cover a distance of 385 km.

Finding the Number of Hours

Let's assume that the car takes n hours to cover a distance of 385 km. We can set up an equation based on the distances traveled in each hour:

40 + 45 + 50 + ... + (40 + 5(n-1)) = 385

We can simplify this equation by using the formula for the sum of an arithmetic series:

n/2 × (40 + 385) = 385

Simplifying further, we get:

n/2 × 425 = 385

Multiplying both sides by 2, we get:

n × 425 = 770

Dividing both sides by 425, we get:

n = 770/425

n = 1.8

However, this is not the correct answer, as the car cannot take a fraction of an hour to cover a distance. We need to round up to the nearest whole number.

Rounding Up to the Nearest Whole Number

Since the car cannot take a fraction of an hour to cover a distance, we need to round up to the nearest whole number. In this case, we round up to 2 hours.

However, this is still not the correct answer, as the car takes more than 2 hours to cover a distance of 385 km. We need to continue adding hours until we reach a total distance of 385 km.

Continuing to Add Hours

Let's continue to add hours until we reach a total distance of 385 km:

Hour 1: distance = 40 km
Hour 2: distance = 40 + 45 = 85 km
Hour 3: distance = 85 + 50 = 135 km
Hour 4: distance = 135 + 55 = 190 km
Hour 5: distance = 190 + 60 = 250 km
Hour 6: distance = 250 + 65 = 315 km
Hour 7: distance = 315 + 70 = 385 km

We can see that the car takes 7 hours to cover a distance of 385 km.

Conclusion

To solve this problem, we need to use a different approach than the formula for distance under uniform acceleration. We need to break down the problem into smaller time intervals, each of 1 hour, and calculate the distance traveled in each hour. We can then sum up the distances traveled in each hour to find the total distance traveled. Finally, we need to round up to the nearest whole number to find the number of hours it takes to cover a distance of 385 km.

The correct answer is D. 7 h.

Understanding the Problem

Formula for Distance Under Uniform Acceleration

The formula for calculating distance traveled under uniform acceleration is given by:

d = ut + (1/2)at^2

where:

d is the distance traveled
u is the initial speed
t is the time
a is the acceleration

However, in this case, the acceleration is not constant, but rather increases by 5 km/h every hour. To solve this problem, we need to use a different approach.

Calculating Distance Traveled

We can calculate the distance traveled in each hour as follows:

Hour 1: distance = speed × time = 40 km/h × 1 h = 40 km
Hour 2: distance = speed × time = 45 km/h × 1 h = 45 km
Hour 3: distance = speed × time = 50 km/h × 1 h = 50 km
...
Hour n: distance = speed × time = (40 + 5(n-1)) km/h × 1 h = (40 + 5(n-1)) km

We can see that the distance traveled in each hour increases by 5 km.

Finding the Total Distance Traveled

To find the total distance traveled, we need to sum up the distances traveled in each hour. We can do this by using the formula for the sum of an arithmetic series:

d = n/2 × (a + l)

where:

d is the total distance traveled
n is the number of hours
a is the initial distance (40 km)
l is the last distance (385 km)

However, we need to find the value of n, which is the number of hours it takes to cover a distance of 385 km.

Finding the Number of Hours

Let's assume that the car takes n hours to cover a distance of 385 km. We can set up an equation based on the distances traveled in each hour:

40 + 45 + 50 + ... + (40 + 5(n-1)) = 385

We can simplify this equation by using the formula for the sum of an arithmetic series:

n/2 × (40 + 385) = 385

Simplifying further, we get:

n/2 × 425 = 385

Multiplying both sides by 2, we get:

n × 425 = 770

Dividing both sides by 425, we get:

n = 770/425

n = 1.8

However, this is not the correct answer, as the car cannot take a fraction of an hour to cover a distance. We need to round up to the nearest whole number.

Rounding Up to the Nearest Whole Number

Since the car cannot take a fraction of an hour to cover a distance, we need to round up to the nearest whole number. In this case, we round up to 2 hours.

However, this is still not the correct answer, as the car takes more than 2 hours to cover a distance of 385 km. We need to continue adding hours until we reach a total distance of 385 km.

Continuing to Add Hours

Let's continue to add hours until we reach a total distance of 385 km:

Hour 1: distance = 40 km
Hour 2: distance = 40 + 45 = 85 km
Hour 3: distance = 85 + 50 = 135 km
Hour 4: distance = 135 + 55 = 190 km
Hour 5: distance = 190 + 60 = 250 km
Hour 6: distance = 250 + 65 = 315 km
Hour 7: distance = 315 + 70 = 385 km

We can see that the car takes 7 hours to cover a distance of 385 km.

Conclusion

The correct answer is D. 7 h.

Q&A

Q: What is the initial speed of the car?

A: The initial speed of the car is 40 km/h.

Q: How much does the car's speed increase every hour?

A: The car's speed increases by 5 km/h every hour.

Q: What is the total distance traveled by the car?

A: The total distance traveled by the car is 385 km.

Q: How many hours does it take for the car to cover a distance of 385 km?

A: It takes the car 7 hours to cover a distance of 385 km.

Q: What is the formula for calculating distance traveled under uniform acceleration?

A: The formula for calculating distance traveled under uniform acceleration is given by:

d = ut + (1/2)at^2

where:

d is the distance traveled
u is the initial speed
t is the time
a is the acceleration

Q: Why can't we use the formula for distance under uniform acceleration to solve this problem?

A: We can't use the formula for distance under uniform acceleration to solve this problem because the acceleration is not constant, but rather increases by 5 km/h every hour.

Q: How do we calculate the distance traveled in each hour?

A: We calculate the distance traveled in each hour by multiplying the speed by the time.

Q: How do we find the total distance traveled?

A: We find the total distance traveled by summing up the distances traveled in each hour.

Q: Why do we need to round up to the nearest whole number?

A: We need to round up to the nearest whole number because the car cannot take a fraction of an hour to cover a distance.

Q: What is the correct answer?

A: The correct answer is D. 7 h.