A 1400-kg sports car accelerates from rest to 95 km/h in 7.4 s. What is the average power delivered

Summary

This video script explains how to calculate the average power delivered by a sports car's engine during acceleration. Starting from rest, the car reaches 95 km/h in 7.4 seconds. The process involves converting speed to meters per second, calculating kinetic energy change, and then dividing by time to find power in watts. The script also demonstrates converting watts to horsepower, resulting in an estimated 88 horsepower for the car's engine.

Takeaways

• 🏎️ The problem involves calculating the average power of a 1400 kg sports car accelerating from rest to 95 km/h in 7.4 seconds.
• 🔧 The formula for power is work over time, which in this context is the change in kinetic energy over the time taken to accelerate.
• ⏱️ The given time for acceleration is 7.4 seconds.
• 🚫 The script notes that direct calculation of work is not possible due to the lack of force data.
• 🔄 The work done is equated to the change in kinetic energy, which is calculated using the formula for kinetic energy: \( \frac{1}{2}mv^2 \).
• 📐 The initial velocity (v1) is zero since the car starts from rest.
• 🔢 The final velocity (v2) must be converted from 95 km/h to meters per second, resulting in approximately 26.389 m/s.
• 📉 The change in kinetic energy is found by subtracting the initial kinetic energy (which is zero) from the final kinetic energy.
• 📈 The mass of the car is given as 1400 kg, which is used in the kinetic energy calculation.
• 📊 The calculated change in kinetic energy is approximately 487,465.52 joules.
• ⏳ The average power is then found by dividing the change in kinetic energy by the time taken, resulting in 65,873.71 watts.
• 🐎 An estimation is made to convert the power from watts to horsepower, yielding approximately 88 horsepower.

Q & A

What is the given mass of the sports car in the problem?

The given mass of the sports car is 1400 kilograms.

What is the final velocity of the sports car?

The final velocity of the sports car is 95 kilometers per hour.

How long does it take for the sports car to reach its final velocity?

It takes 7.4 seconds for the sports car to reach its final velocity.

What is the formula for power used in this problem?

The formula for power used in this problem is power = work / time.

How is work related to kinetic energy in this context?

Work is equal to the change in kinetic energy in this context.

What is the formula for kinetic energy?

The formula for kinetic energy is KE = 1/2 * m * v^2.

How do you convert the final velocity from kilometers per hour to meters per second?

To convert 95 kilometers per hour to meters per second, multiply by 1000 to convert kilometers to meters and then divide by 3600 to convert hours to seconds, resulting in 26.389 meters per second.

What is the change in kinetic energy of the sports car?

The change in kinetic energy of the sports car is 487,465.5247 joules.

How do you calculate the average power delivered by the engine?

The average power is calculated by dividing the change in kinetic energy by the time, resulting in 65,873.71 watts.

How do you convert the power from watts to horsepower?

To convert power from watts to horsepower, divide the power in watts by 746. In this case, 65,873.71 watts is approximately 88 horsepower.

Outlines

00:00

🚗 Calculating Average Power for a Sports Car

This paragraph discusses the problem of calculating the average power delivered by the engine of a 1400-kilogram sports car that accelerates from rest to 95 kilometers per hour in 7.4 seconds. The approach involves using the formula for power, which is work over time. The work is determined by the change in kinetic energy since force and distance are not provided. The change in kinetic energy is calculated using the formula 0.5 * m * (v2^2 - v1^2), where v2 is the final velocity (converted from 95 km/h to 26.389 m/s) and v1 is the initial velocity (0 m/s). Plugging in the values and performing the calculations gives the work done, which is then divided by the time (7.4 seconds) to find the average power in watts.

05:02

🔧 Converting Power to Horsepower

This paragraph focuses on converting the calculated power from watts to horsepower. The previously calculated power in watts (65,873.71 W) is approximated to 66,000 W for ease of conversion. Knowing that 1 horsepower equals 746 watts, the power in watts is divided by 746 to get the power in horsepower, resulting in approximately 88 horsepower. The paragraph also notes that for more precision, the exact watt value should be used instead of the rounded number.

Keywords

💡Power

Power in the context of this video refers to the rate at which work is done or energy is transferred over time. It is a key concept in understanding the car's engine performance. The script uses the formula 'power = work over time' to calculate the average power delivered by the engine as the car accelerates from rest to 95 km/h in 7.4 seconds.

💡Work

Work, in physics, is the product of force and the distance over which it acts in the direction of the force. In the script, work is related to the change in kinetic energy of the car, which is used to calculate the power output of the engine since the direct measurement of force is not provided.

💡Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula '0.5 * m * v^2', where 'm' is mass and 'v' is velocity. The script explains that the work done on the car can be equated to the change in its kinetic energy from rest to its final velocity.

💡Acceleration

Acceleration is the rate of change of velocity of an object. The video script describes a sports car accelerating from rest to 95 km/h, which is a measure of how quickly the car's velocity changes over the 7.4-second period.

💡Velocity

Velocity is the speed of an object in a specific direction. The script mentions two velocities: the initial velocity at rest (0 m/s) and the final velocity of 95 km/h, which is converted to meters per second for calculations.

💡Mass

Mass is a measure of the amount of matter in an object. In the script, the mass of the sports car is given as 1400 kilograms, which is a crucial factor in calculating the change in kinetic energy and, subsequently, the power output.

💡Formula

A formula in this context is a mathematical equation used to calculate a specific value. The script uses several formulas, including the formula for kinetic energy and the relationship between power, work, and time, to solve for the average power delivered by the engine.

💡Watt

A watt is a unit of power in the International System of Units (SI), equal to one joule per second. The script calculates the power in watts by dividing the change in kinetic energy by the time taken for acceleration.

💡Horsepower

Horsepower is a unit of measurement of power, often used in the automotive industry. The script provides a conversion from watts to horsepower, showing that 1 horsepower is equivalent to 746 watts, to give an estimated power output of the car's engine in a more familiar unit.

💡Conversion

Conversion in this script refers to changing the units of measurement, such as from kilometers per hour to meters per second, or from watts to horsepower. This is necessary to ensure that all values are in compatible units for the calculations to be accurate.

💡Estimation

Estimation is the process of approximating a value or quantity. The script mentions rounding numbers for ease of calculation and acknowledges that the final horsepower figure is an estimate, not an exact value.

Highlights

The problem involves calculating the average power of a sports car accelerating from rest to 95 km/h in 7.4 seconds.

Power is calculated as work over time, which is the formula used to solve the problem.

Work is equated to the change in kinetic energy, as the force is not given.

Kinetic energy formula is given as \( \frac{1}{2}mv^2 \).

The change in kinetic energy is calculated by subtracting the initial kinetic energy from the final kinetic energy.

Initial velocity is 0 m/s since the car starts from rest.

Final velocity is converted from 95 km/h to 26.389 m/s.

The change in kinetic energy is calculated using the mass of the car (1400 kg) and the final velocity squared.

The calculated change in kinetic energy is 487465.524 Joules.

Power is then found by dividing the change in kinetic energy by the time taken (7.4 seconds).

The average power delivered by the engine is calculated to be 65873.71 Watts.

To convert the power to horsepower, 66000 Watts is used for easier calculation.

One horsepower is equivalent to 746 Watts.

The estimated horsepower is approximately 88, based on the rounded Watts value.

An exact horsepower calculation would use the precise value of 65873.71 Watts.

The solution process emphasizes the application of physics formulas to real-world problems.

The importance of unit conversion in solving physics problems is highlighted.

The transcript demonstrates a step-by-step approach to solving for power using kinetic energy.

Transcripts

in this problem we're told a 1400

kilogram sports car accelerates from

rest to 95 kilometers per hour in 7.4

seconds

what is the average power delivered by

the engine

so in order to solve this problem you

need to know this formula you need to

know that power

is going to be equal to work over time

and so keep in mind what we're trying to

solve for we're trying to solve for

power

so we just got to take the work and then

divide it by the time and so we do know

the time in this case

it's going to be uh 7.4 seconds

right so 7.4 seconds and we just got to

take the work and divide it by that

but what is the work gonna be so in this

problem we can't really solve for work

right because work equals force times

distance and we don't really know a

force they don't give us that

but what you can do is say you need to

know that work is equal to the change in

kinetic energy

right and if that's the case what we can

do is just solve for the change in

kinetic energy

so the formula for kinetic energy is one

half mv squared

right so notice that they give us

velocity so this is what they want you

to do

and so how do we find the change in

kinetic energy though

so essentially the change in kinetic

energy is just going to be one half

mv two squared right

minus one half mv one squared you're

just taking the kinetic

energy at the end right wherever your

velocity is

in this case they're saying at 95

kilometers per hour and then

this one is going to be in the beginning

so essentially just take it at the end

minus it at the beginning so we can

rewrite this though as just

m one half m and then i'm just factoring

that up

out of both so it's just v two squared

minus v one squared

so essentially this is gonna be the

change in kinetic energy and we can just

put this up here and then divided by 7.4

and that's going to be our answer

so let's go ahead and solve this first

let's find the

velocities so v sub 1 and v sub 2

which is your velocity in the beginning

velocity at the end so they tell us we

start from rest

so if we start from rest that means our

velocity is zero meters per second

and then uh

keep in mind also for this one v sub two

they give us that it's 95 kilometers per

hour

but when we solve this it has to be in

meters per second so we have to convert

this into meters per second

so let's go ahead and do that so 95

kilometers per hour

we know that there's 1 000 meters for

every one kilometer

that'll cancel the kilometers right

multiply by a thousand and then that for

every one hour there's 3 600 seconds

right that'll get rid of the hour and

then we just have meters over seconds

so essentially just do 95 times a

thousand right which is just 95 000 and

then divide that by

3 600. if you go ahead and do that

you're going to get

26.388 and so on i'm just going to round

it to 26.389

and so meters per second so this is

going to be

your velocity now

and so all we can do is just plug it in

now and then we'll be able to solve

right

so the change in kinetic energy is one

half

times your mass and so they did tell us

in the beginning the mass

of our sports car is 1400 kilograms

so 1400 and then v2 is

uh 26.389

squared minus v sub one which is just

zero zero squared is just zero

so it's essentially just one half times

fourteen hundred times twenty six point

three eight nine squared

so if you go ahead and do that you're

gonna get that

equals so 700 times 26.389

squared you're gonna get that the change

in kinetic energy is equal to

four eight seven four six

five so four hundred eighty seven

thousand four hundred sixty five

point five two four seven

you can go as far as you want whatever

you wanna do uh essentially

so this is gonna be your change in

kinetic energy and we measure it in

joules so

right because energy's measured in

joules so i can't really fit the j but

just know what's in joules so if we want

to solve for the power now

right change in kinetic energy is equal

to the work so we just take the work

divided by time

so just take this number for 487

465 and then divided by

7.4 so p

is equal to 487

465.524

divide that by 7.4

and so if you go ahead and do this

you're going to get that equals so

if you if you do this six five it's

equal to six

five eight seven three point seven one

so sixty five thousand eight hundred

seventy three point

seven one and so keep in mind this is

gonna be in watts

so when you take joules and divided by

time you get watts

so this is 65 873.71 watts

but uh you can also find it in

horsepower so i'm going to show you how

to do that

so i'm going to erase everything on

screen so if you want to write it down

write it down

but i'm just going to need this number

in order to convert it to horsepower

so it's 65

873

0.7 i'm just gonna i'm just gonna round

this so

essentially i'm just gonna make it 66

000 i'm just rounding up

just because it's going to make it

easier when converting if your teacher

wants you to do the exact version just

make sure you use this number when

converting

so 66 000 and then i'm going to

convert it or 66 000 watts and so if you

want to take watts and convert it into

horsepower

we know that one horsepower is equal to

746 watts

if that's the case right this is in

watts

so 66 000 watts

multiply that by one horsepower and then

over 746

watts so right because if we have

something

in watts and we just divide it by 746

that's going to give it horsepower

so 66 000 divided by 746

if you go ahead and do that you're going

to get 88.4718 and so on

i'm just going to round to the whole

number so it's going to be about 88

horsepower

keep in mind we did use an estimated

version when calculating the horsepower

so it's not as exact but

if you want it exact just use this

number right here which was

to the decimal point right in watts but

yeah so 88 horsepower

is our estimated answer