A copper block with a mass 1.82 kg is given an initial speed of 3.5 m/s on a rough horizontal surface. Because of friction, the block finally comes to rest. Assume that 77.4 % of the initial kinetic energy is absorbed by the block in form of heat, what is the change in temperature (in - do not enter units) of the block? Data: cCu=387

To find the change in temperature of the copper block, we need to first calculate the initial kinetic energy of the block, and then determine the amount of energy absorbed by the block in the form of heat.

1. Calculate initial kinetic energy:
The formula for kinetic energy is: KE = (1/2) * mass * velocity^2
Given:
Mass of the copper block (m) = 1.82 kg
Initial speed (v) = 3.5 m/s

Using the formula, we can calculate the initial kinetic energy (KE_initial):
KE_initial = (1/2) * m * v^2

2. Calculate the energy absorbed:
Given:
Percent of energy absorbed = 77.4%

To find the energy absorbed by the block in the form of heat, we need to multiply the initial kinetic energy by the percentage (expressed as a decimal):
Energy_absorbed = KE_initial * (77.4/100)

3. Calculate the change in temperature:
The formula for the change in temperature (ΔT) using the specific heat capacity (c) is:
Change in temperature = Energy_absorbed / (mass * specific heat capacity)

Given:
Specific heat capacity of copper (cCu) = 387

Using the formula, we can calculate the change in temperature (ΔT):
ΔT = Energy_absorbed / (m * cCu)

Now we can substitute the values into the formula to find the change in temperature. Remember to convert the given percentage to a decimal before calculating:

Energy_absorbed = KE_initial * (77.4/100)
ΔT = Energy_absorbed / (m * cCu)

To find the change in temperature of the copper block, we need to use the formula:

Q = mcΔT

Where:
Q is the heat absorbed by the copper block
m is the mass of the copper block
c is the specific heat capacity of copper
ΔT is the change in temperature

Given:
m = 1.82 kg (mass of the copper block)
cCu = 387 (specific heat capacity of copper)
KE_initial = 0.5 * m * (v^2) (initial kinetic energy of the block)
v = 3.5 m/s (initial speed of the block)

First, let's find the initial kinetic energy (KE_initial) of the copper block:
KE_initial = 0.5 * 1.82 kg * (3.5^2)
KE_initial = 21.6725 J

Next, let's calculate the heat absorbed by the copper block (Q) using the given information that 77.4% of the initial kinetic energy is absorbed:
Q = 0.774 * KE_initial
Q = 0.774 * 21.6725 J
Q ≈ 16.7798 J

Finally, we can find the change in temperature (ΔT) using the formula mentioned earlier:
Q = mcΔT
ΔT = Q / (mc)
ΔT = 16.7798 J / (1.82 kg * 387)
ΔT ≈ 0.0241 °C

Therefore, the change in temperature of the copper block is approximately 0.0241 °C.