Question

b. Andrea has a particularly old bike, and she has not lubricated the chain orgears. This causes friction inside the gears as she rides. She begins coasting onflat ground at a speed of 10 m/s, and after a few seconds is only going 8 m/s.How much energy has been converted to heat? (1 point) (The combined mass is 50kg)c. Andrea is going 10 m/s toward a hill. She coasts up the hill without pedaling. Iffriction causes 10% of her energy to be converted to heat inside the gears, howhigh up the hill will she be able to coast? (2 points)

Answer 1

b.

The energy converted in heat is the change in kinetic energy, that is, the kinetic energy lost is the same as the heat.

The kinetic energy is given by:

`K=(1)/(2)mv^2`

The change in kinetic energy is given by:

`K_f-K_0=(1)/(2)mv^2_f-(1)/(2)mv^2_0`

Plugging the values given we have that:

`\begin{gathered} K_f-K_0=(1)/(2)(50)(8)^2-(1)/(2)(50)(10)^2 \\ =-900 \end{gathered}`

Hence Andrea has lost 900 J of energy and therefore 900 J is converted to heat.

c.

We know that the energy is conserved, in this case this means that the initial kinetic energy has to be equal to the energy lost by heat in the gears and the potential energy gained by climbing up the hill, that is:

`K=Q+U`

where K is the kinetic energy, Q is the heat and U is the potential energy.

We know that 10% of her energy is converted in heat, this means that:

`Q=0.1K`

and hece we have:

`\begin{gathered} K=0.1K+U \\ U=0.9K \end{gathered}`

the potential energy is given by:

`U=mgh`

then we have that:

`mgh=0.9((1)/(2)mv^2)`

Plugging the values given and solving for h we have:

`\begin{gathered} (50)(9.8)h=0.9((1)/(2)(50)(10^2)) \\ 490h=2250 \\ h=(2250)/(490) \\ h=4.59 \end{gathered}`

Therefore Andrea will be able to climp up to 4.59 meters