Just as a car tops a 51 meter high hill with a speed of 85 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?
v =85 km/h= 23.61 m/s.
mv^2/2+mgh = mgH.
H = h +v^2/2g =51 +28.44 =79.44 m.
"solutio"n v=85km/h v=85*1000/60*60 v=85000/3600 v=23.61m/s H=h+v*v/2g =51+23.61*23.61/2*9.8 =51+557.4/19.6 =51+28.44 H=79.44m.
To find out how high the car will coast up the next hill, we need to analyze the energy conservation of the car as it goes up the first hill.
First, let's convert the car's speed from kilometers per hour to meters per second. We know that 1 kilometer is equal to 1000 meters and 1 hour is equal to 3600 seconds.
85 km/h * (1000 m/km) / (3600 s/h) = 23.6 m/s
Next, let's calculate the initial kinetic energy (K1) of the car at the top of the first hill. The kinetic energy is given by the equation:
K1 = (1/2) * m * v^2
where m is the mass of the car and v is its velocity.
The mass of the car is not given, but since it is not relevant for our calculations, we can disregard it in this scenario.
K1 = (1/2) * v^2
K1 = (1/2) * (23.6 m/s)^2
K1 = 278.45 J (to 2 decimal places)
The initial kinetic energy (K1) of the car at the top of the hill is 278.45 Joules.
As the car coasts up the next hill, it will gradually lose kinetic energy and convert it into potential energy (the energy associated with height). To find out how high the car will coast up the next hill, we can equate the initial kinetic energy (K1) to the potential energy gained (PE). The potential energy is given by the equation:
PE = m * g * h
where m is the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the hill.
Since the mass of the car doesn't affect the height it will coast, we can disregard it again in this scenario.
K1 = m * g * h
h = K1 / (m * g)
Now, we need to find the mass of the car. Given that we only have the car's speed and height information, we can compute the mass by dividing the potential energy gained in ascending the first hill by the acceleration due to gravity.
The potential energy gained (PE) for climbing the first hill is equal to the car's initial kinetic energy (K1):
PE = K1 = 278.45 J
Now we have:
h = K1 / (m * g)
h = 278.45 J / (m * 9.8 m/s^2)
h = 28.41 m (to 2 decimal places)
Therefore, the car will coast up the next hill to a height of approximately 28 meters (to the nearest meter).