Velocity and Acceleration Problems and Solutions

The velocity of an object is equal to the distance the object travels divided by the time it takes to travel that distance.

Problem:

A car travels 2 miles in 4 minutes. Convert this velocity to miles per hour. How many miles does the car travel in 6 hours?

Solution:

One hour equals sixty minutes.

To solve this problem, set up a ratio in fraction form

(2 miles)/(4 minutes) = (x miles)/(60 minutes)

Now our solution involves algebraic manipulations

Multiply both sides of the equation by 60 minutes

60 minutes * (2 miles)/(4 minutes) = 60 minutes * (x miles)/(60 minutes)

60 minutes * (2 miles)/(4 minutes) = x miles

(60 minutes)/(4 minutes) = 15

(15 * 2 miles)/(1) = 30 miles

The answer is 30 miles per hour

Alternative Solution:

(2 miles)/(4 minutes) = (x miles)/(60 minutes)

Since 4 minutes * 15 = 60 minutes, multiply numerator and denominator of left side of equation by 15. (This is equivalent to multiplying by one and therefore the right side of the equation is not affected.)

(2 miles * 15)/(4 minutes * 15) = (30 miles)/(60 minutes) = (30 miles)/(hour)

In six hours travels ...

miles per hour * hours = miles traveled

30 miles per hour * 6 hours = 180 miles

The car travels 180 miles in six hours.

ACCELERATION

Acceleration is the increase in velocity and is measured as follows:

acceleration = (change of velocity)/(time) = (distance/time)/(time)

acceleration = (distance/time)/(time)

Problem:

A car is accelerating from a standstill at 6 miles per hour per hour. How fast is the car going in two hours? What is the average velocity? How far does the car travel in two hours?

Solution:

acceleration * time = velocity

6 miles per hour per hour * 2 hours = 12 miles per hour

So at the end of 2 hours the velocity is 12 miles per hour. However, this velocity has been increasing linearly since the car started moving. If we use the velocity 12 miles per hour to find the distance traveled, the distance found would be the distance traveled by a car traveling at 12 miles per hour for two hours. But the car was not traveling at a constant velocity. The car was accelerating. Hence we must use the average velocity to calculate the distance traveled.

average velocity = (acceleration * time)/2

acceleration = 6 miles per hour per hour

time = 2 hours

average velocity = 6 miles per hour per hour * 2 hours/2 = 6 miles per hour

distance = (average velocity * time)

distance = 6 miles per hour * 2 hours = 12 miles

In cases where the initial velocity is greater than zero, we would use the equation

distance = initial velocity * time + (acceleration * time)/2

Problem:

A car traveling at 60 miles per hour decelerates at 10 miles per hour per hour. How long before the car comes to a complete stop?

Solution:

Acceleration is the change in velocity per unit of time.

The velocity of the car decreases by ten miles per hour each hour

First hour 50 miles per hour.

Second hour: 40 miles per hour.

Third hour: 30 miles per hour.

Fourth hour: 20 miles per hour.

Fifth hour: 10 miles per hour.

Sixth hour: 0 miles per hour.

proof: acceleration = (velocity - original velocity)/time = -60/6 = -10 miles per hour per hour

An alternative solution for the time to stop is to use the formula

velocity = original velocity + acceleration * time

to find the time the car took to stop

as long as we remember that

velocity = 0 miles per hour

and

original velocity = 60 miles per hour

and acceleration = -10 miles per hour per hour.

velocity = original velocity + acceleration * time

0 = 60 mph + (-10)*time

-60 mph = (-10)*time

t = (-60)/(-10) = 6 hours.

The average velocity is

(60 miles per hour - 0 miles per hour)/2 = 30 miles per hour

distance traveled = average velocity * time

30 miles per hour * 6 hours = 180 miles

The car traveled 180 miles

References:
New Second Algebra
Library of Congress Category Card Number 62-7240