Question

The engine in an imaginary sports car can provide constant power to the wheels over a range of speeds from 0 to 70 miles per hour (mph). At full power, the car can accelerate from zero to 29.0mph in time 1.50s .

Part A

At full power, how long would it take for the car to accelerate from 0 to 58.0mph ? Neglect friction and air resistance. THE ANSWER IS "6 seconds" according the mastering physics.. BUT WHY???? Please show how you derive that answer...

Part B

A more realistic car would cause the wheels to spin in a manner that would result in the ground pushing it forward with a constant force (in contrast to the constant power in Part A). If such a sports car went from zero to 29.0mph in time 1.50s , how long would it take to go from zero to 58.0mph ?

Answer 1

Answer:

A) 6 secs

B) 3 secs

Explanation:

A) CONSTANT POWER

At full power, we have:

Initial velocity = 0mph

Final velocity = 29mph

t = 1.5secs

We are told that the power is constant. Hence, we first find the power applied by the engine of the car:

P = E/t

Where P = power

E = Energy

t = time (in seconds)

Kinetic Energy (E) is given as:

E = 0.5*m*v²

We are not given the mass, so:

P = (0.5*m*v²)/t

P = (0.5 * m * 29²)/1.5

P = 280m J/s

This power is constant but now the final velocity is 59 mph. We can find time by making t subject of the power formula:

P = (0.5*m*v²)/t

t = (0.5*m*v²)/P

t = (0.5*m*58²)/(280m)

t = 6 secs.

B) CONSTANT FORCE

In this case, force is constant:

F = ma = constant

Where a = acceleration.

We can find acceleration, a, by using one of the equations of motion:

a = (v - u)/t

In hours,

t = 1.5/3600 = 0.0004167 hr

a = (29 - 0)/(0.0004167)

a = 69600 mph²

Since the force is constant and mass is constant, that means the acceleration is also constant. Hence, when the final velocity is 58mph, t will be:

a = (v - u)/t

=> t = (v - u)/a

u = 0mph

=> t = 58/69600

t = 0.000833 hr = 3secs

Answer 2

A) The time it would take for the car to accelerate from 0 to 58.0mph at full power is; t = 6 seconds

B) The time it would take would it take to go from zero to 58.0mph at constant force is; t = 3 seconds

Formula for Kinetic energy(workdone) is;

KE = ¹/₂mv²

Now, the car can accelerate from 0 to 29 mph(12.9642 m/s) in 1.5 s. Thus;

KE₁ = ¹/₂ × m × 12.9642²

KE₁ = 84.035m J

Now, formula for power is;

P = work done/time taken

Thus; P₁ = 84.035m/1.5

P₁ = 56.02267m W

Since the full power is  56.02267m W, then the time it will take the car to accelerate to 58 mph(25.9283 m/s) is gotten from;

t = work done/power

Work done here is; KE = ¹/₂ × m × 25.9283²

KE = 336.13837m J

Thus; t = 336.13837m/56.02267m

t = 6 seconds

B) formula for acceleration is;

a = (v - u)/t

a = (12.9642 - 0)/1.5

a = 8.6428 m/s²

Since force is constant, then time is gotten from;

t = (v - u)/a

t = (25.9283 - 0)/8.6428

t = 3 seconds

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