Question

A woman drives 10 miles, accelerating uniformly from rest to 60 mph. Graph her velocity versus time. How long does it take for her lo reach 60 mph?

Answer 1

To graph the velocity versus time for the woman's drive, we can use the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken. The time it takes for her to reach 60 mph is 20 minutes.

Step-by-step explanation:

To graph the velocity versus time for the woman's drive, we can use the equation:

v = u + at

Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.

In this case, the woman starts from rest, so u = 0 mph. The final velocity is 60 mph, and the distance traveled is 10 miles. We are looking for the time it takes for her to reach 60 mph, so we can rearrange the equation to solve for t:

t = (v - u)/a

Substituting the given values:

t = (60 mph - 0 mph)/(a)

To find the acceleration, we can use the kinematic equation:

v^2 = u^2 + 2as

Where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance traveled. Rearranging the equation to solve for a:

a = (v^2 - u^2)/(2s)

Substituting the given values:

a = (60 mph^2 - 0 mph^2)/(2 * 10 miles)

Converting the units to mph/miles:

a = 3 mph/miles

Now we can substitute the acceleration into the equation for time:

t = (60 mph - 0 mph)/(3 mph/miles)

Calculating the time:

t = 20 minutes

Answer 2

When a car accelerates uniformly, its velocity increases at a constant rate. To graph the velocity versus time, we need to know the acceleration she's undergoing.

The formula to calculate the time it takes to reach a certain velocity under constant acceleration is:

`\[ v = u + at \]`

Where:

`\( v \) = final velocity (60 mph in this case)`

`\( u \) = initial velocity (0 mph as she starts from rest)`

`\( a \) = acceleration`

`\( t \) = time`

To find acceleration, we can use another formula:

`\[ v = u + at \]`

`\[ 60 \, \text{mph} = 0 \, \text{mph} + a * t \]`

Given that the initial velocity is 0 mph, the final velocity is 60 mph, and the distance covered is 10 miles, let's convert the velocities to consistent units of miles per hour (mph) and time to hours to maintain consistency:

`\[ \text{Distance} = \text{Average velocity} * \text{Time} \]`

`\[ \text{Time} = \frac{\text{Distance}}{\text{Average velocity}} \]`

So, the average velocity can be calculated as:

`\[ \text{Average velocity} = \frac{\text{Initial velocity} + \text{Final velocity}}{2} \]`

`\[ \text{Average velocity} = \frac{0 \, \text{mph} + 60 \, \text{mph}}{2} = 30 \, \text{mph} \]`

Let's find the time taken:

`\[ \text{Time} = \frac{10 \, \text{miles}}{30 \, \text{mph}} = (1)/(3) \, \text{hour} = 20 \, \text{minutes} \]`

So, she takes
`\( (1)/(3) \)` of an hour to reach 60 mph.

As for the graph of her velocity versus time, during this constant acceleration, the graph will be a straight line. The slope of this line represents the constant rate of acceleration. Since she starts from rest and accelerates uniformly to reach 60 mph in 20 minutes, the graph will show a linear increase from 0 mph to 60 mph over this time duration.