Just as a car tops a 49 meter high hill with a speed of 77 km/h it runs out of gas and coasts from there, without friction or drag. How high, to the nearest meter, will the car coast up the next hill?
plzzz help!
77 km/h = 21.4 m/s
m•g•h+m•v^2/2 = m•g•H
H = h+v^2/2•g = 49 +(21.4)^2/2•9.8 =72.4 m
To determine how high the car will coast up the next hill after running out of gas, we need to consider the laws of conservation of energy.
The car's initial energy at the top of the first hill is in the form of both kinetic energy (KE) and gravitational potential energy (PE). As the car coasts up the next hill, it will lose kinetic energy but gain potential energy.
1. Convert the speed from km/h to m/s:
Speed = 77 km/h * (1000 m/km) / (3600 s/h) = 21.39 m/s
2. Calculate the initial kinetic energy at the top of the first hill:
KE = (1/2) * mass * speed^2
Since the mass of the car isn't given, we can cancel it out by using ratios.
Let's assume the mass is the same as an average car, which is around 1500 kg.
KE = (1/2) * 1500 kg * (21.39 m/s)^2
3. Calculate the initial potential energy at the top of the first hill:
PE = mass * gravity * height
Assuming the mass is still 1500 kg, gravity is approximately 9.8 m/s^2, and the height is 49 meters.
PE = 1500 kg * 9.8 m/s^2 * 49 m
4. The total initial energy is the sum of kinetic and potential energies:
Total initial energy = KE + PE
5. As the car coasts up the next hill, it will lose kinetic energy due to the absence of any external force. This means the loss in kinetic energy will be equal to the gain in potential energy.
6. Set up an equation equating the loss in kinetic energy and the gain in potential energy:
Loss in KE = Gain in PE
7. Solve the equation for the height of the next hill (gain in potential energy):
Gain in PE = Total initial energy - Loss in KE
Gain in PE = Total initial energy
Now, let's calculate the height to the nearest meter:
Substituting the values obtained in steps 2 and 3 into step 4, we get:
Total initial energy = KE + PE
Total initial energy = (1/2) * 1500 kg * (21.39 m/s)^2 + 1500 kg * 9.8 m/s^2 * 49 m
Calculate this total initial energy.
Once you have the value for the total initial energy, substitute it into step 7 to find the height of the next hill (gain in potential energy).
Note: Remember to convert the result back to meters and round to the nearest meter.
To solve this problem, we can apply the principle of conservation of energy. At the top of the first hill, the car has a certain amount of mechanical energy, which consists of kinetic energy (due to its speed) and potential energy (due to its height above the ground). As the car coasts up the next hill, its kinetic energy will be converted into potential energy.
Let's break down the problem and calculate the answer step by step:
Step 1: Convert the speed of the car from km/h to m/s.
To convert km/h to m/s, divide the speed by 3.6:
77 km/h ÷ 3.6 ≈ 21.39 m/s (rounded to two decimal places)
Step 2: Calculate the initial potential energy at the top of the first hill.
The potential energy (PE) can be calculated using the formula: PE = m * g * h, where m is the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the hill.
Since the problem doesn't provide the mass of the car, we can cancel it out by comparing the heights of the two hills.
Step 3: Calculate the height of the second hill.
Since there is no friction or drag, the total mechanical energy of the car (kinetic energy + potential energy) is conserved. Therefore, the initial potential energy at the top of the first hill is equal to the potential energy at the top of the second hill.
Step 4: Solve for the height of the second hill.
Using the formula for potential energy, PE = m * g * h, and rearranging the formula to solve for the height (h), we have h = PE / (m * g).
Since the mass and gravity factors cancel out when comparing the heights of the two hills, we only need to consider the ratio between their heights.
Step 5: Calculate the height ratio between the hills.
Since the initial potential energy at the top of the first hill is equal to the potential energy at the top of the second hill, the height ratio can be found by dividing the initial height by the acceleration due to gravity (h₁ / g).
Step 6: Calculate the height of the second hill using the height ratio.
Multiply the height ratio by the acceleration due to gravity (g) to find the height of the second hill.
By following these steps, you can find the height of the second hill to the nearest meter. Note that the exact height of the first hill is not required, only the height ratio between the hills is needed.