A car of mass m traveling at speed v on a horizontal road applies its brakes and comes to rest in time t while traveling a distance of d. The amount of internal energy created is?
(1/2) m v^2
The stopping distance and time don't matter. Initial kinetic energy ends up as heat (internal energy).
To calculate the amount of internal energy created, we need to consider the work done by the friction force acting on the car. When the car applies its brakes, the friction force opposes the motion of the car, causing it to slow down and eventually come to rest.
The work done by a force can be calculated as the product of the force and the distance over which the force acts. In this case, the force is the friction force, and the distance is the distance traveled by the car while braking.
The work done by the friction force is given by the equation:
Work = Force × Distance
Now, let's break this down further. The force of friction can be calculated using Newton's second law:
Force = mass × acceleration
In this case, the acceleration is the deceleration of the car caused by braking. When the car comes to rest, its final velocity is zero, so we can calculate the deceleration using the equation:
v = u + at
where v is the final velocity (zero in this case), u is the initial velocity (given as v), a is the acceleration, and t is the time taken.
Rearranging the equation to solve for the acceleration, we get:
a = (v - u) / t
Simplifying further, we have:
a = (0 - v) / t
a = -v / t
Now, we can substitute the expression for the acceleration back into the equation for the force:
Force = mass × acceleration
Force = m × (-v / t)
Next, we substitute the expression for the force and the distance into the equation for work:
Work = Force × Distance
Work = m × (-v / t) × d
Finally, the amount of internal energy created is equal to the work done by the friction force, so we have:
Internal Energy = m × (-v / t) × d
Therefore, the amount of internal energy created by the car while braking is given by the expression: m × (-v / t) × d.