Question: A car is on a ramp 100m from the bottom of the ramp (angle of incline=30 degrees):
a. If the potential energy of the car is 80902 J, how much does the car weigh?
b. What is the KE half way down the ramp?
c. What speed is the car moving at the bottom of the ramp?
Please indicate which equation you use, what variable corresponds to each number, and how you got the number that goes with the variable.
Thanks!
a. h = 100sin30 = 50m.
PE = mgh = 80,902J,
m*9.8*50 = 80,902,
m = 165.1kg.
F = mg = 165.1kg * 9.8N/kg = 1618N =
Wt. of car in Newtons.
Wt. = 165.1kg / 0.454kg/lb = 364lbs =
Weight of car in lbs.
b. h = (100/2)sin30 = 25m.
V^2 = Vo^2 + 2gd,
V^2 = 0 + 2*9.8*25 = 490,
V = 22.1m/s.
KE = 0.5mV^2 = 0.5 * 165.1 * (22.1)^2 =
40,318J.
c. V^2 = Vo^2 + 2gd,
V^2 = 0 + 2*9.8*50 = 980,
V = 31.3m/s.
a. To find the weight of the car, we can use the equation:
Weight (W) = Potential Energy (PE) / Height (h)
The potential energy of the car is given as 80902 J, and the height (h) of the ramp is 100m. Plugging these values into the equation:
W = 80902 J / 100m
Now, we need to convert J/m to N (Newtons). 1 J/m is equal to 1 N, so:
W = 809.02 N
Therefore, the weight of the car is 809.02 N.
b. To find the kinetic energy (KE) halfway down the ramp, we can use the equation:
KE = (1/2) * mass (m) * velocity (v)^2
However, we are not given the mass of the car. So, we need to find the mass first. We can use the equation:
Potential Energy (PE) = mass (m) * gravitational acceleration (g) * height (h)
The potential energy is given as 80902 J, the height (h) is 100m, and the gravitational acceleration (g) is approximately 9.8 m/s^2. Plugging these values into the equation:
80902 J = m * 9.8 m/s^2 * 100m
Simplifying:
80902 J = 980 m * m/s^2
Dividing both sides by 980 m/s^2:
m = 82.659 m
Now that we have the mass, we can find the KE halfway down the ramp. The distance halfway down the ramp is 100m / 2 = 50m. Plugging in the values:
KE = (1/2) * 82.659 m * v^2
However, we do not have the velocity (v). So, we cannot solve this part of the question without additional information.
c. To find the speed of the car at the bottom of the ramp, we can use the equation:
Potential Energy (PE) = Kinetic Energy (KE) + Weight (W) * Height (h)
The potential energy is given as 80902 J, the weight (W) is 809.02 N, and the height (h) is 100m. Plugging in these values:
80902 J = KE + 809.02 N * 100m
Simplifying:
80902 J = KE + 80902 J
Therefore, the kinetic energy at the bottom of the ramp is 0 J. Since KE = (1/2) * mass (m) * velocity (v)^2, the velocity (v) at the bottom of the ramp is 0 m/s. Thus, the car is not moving at the bottom of the ramp.
To answer these questions, we can use concepts from Newtonian mechanics such as potential energy, kinetic energy, and gravitational force. We will also need to use trigonometry to calculate the component of gravitational force along the ramp.
a. To find the weight of the car, we can use the formula for potential energy:
Potential energy (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 ramp. In this case, the potential energy is given as 80902 J, and the height of the ramp is 100 m.
So, the equation becomes:
80902 J = m * 9.8 m/s² * 100 m.
To find the weight (force of gravity), we can rearrange the equation to solve for the mass (m):
m = 80902 J / (9.8 m/s² * 100 m).
Simplifying the expression, we get:
m = 82.65 kg.
Therefore, the weight of the car is approximately 82.65 kg.
b. To find the kinetic energy (KE) halfway down the ramp, we can use the principle of conservation of mechanical energy. This principle states that the sum of potential energy and kinetic energy remains constant, neglecting any energy losses due to friction or other factors. At the beginning of the ramp, all of the potential energy is stored, and halfway down the ramp, half of that potential energy will be converted into kinetic energy.
Therefore, the kinetic energy halfway down the ramp will be half of the potential energy:
KE = 1/2 * PE.
Substituting the given potential energy of 80902 J into the equation, we get:
KE = 1/2 * 80902 J.
Simplifying the expression, we find:
KE = 40451 J.
Therefore, the kinetic energy halfway down the ramp is approximately 40451 J.
c. To find the speed of the car at the bottom of the ramp, we can use the principle of conservation of mechanical energy again. This time, we will equate the initial potential energy to the sum of kinetic energy and potential energy at the bottom of the ramp.
PE(initial) = KE(final) + PE(final).
The initial potential energy is given as 80902 J, and the final potential energy at the bottom of the ramp is zero because the car is at ground level. Therefore, the equation becomes:
80902 J = KE(final) + 0.
Simplifying the equation, we find:
KE(final) = 80902 J.
Now, we can use the equation for kinetic energy:
KE = 1/2 * m * v²,
where m is the mass of the car and v is its velocity.
Since we already know the mass of the car is 82.65 kg, we can rearrange the equation to solve for the velocity:
v = sqrt(2 * KE / m).
Substituting in the given kinetic energy of 80902 J and the mass of 82.65 kg, we get:
v = sqrt(2 * 80902 J / 82.65 kg).
Evaluating the expression, we find:
v ≈ 56.7 m/s.
Therefore, the car is moving at approximately 56.7 m/s at the bottom of the ramp.