A roller-coaster car has a mass of 500 kg when fully loaded with passengers. At the bottom of a circular dip of radius 40 m (as shown in the figure) the car has a speed of 16 m/s. What is the magnitude of the vertical height above the lowest point, when the speed is zero? Ignore friction.
We can solve this problem using conservation of energy. At the bottom of the dip, all of the car's initial energy is in the form of kinetic energy, since it is moving and has no height. At the point where the speed is zero, all of the car's energy is in the form of potential energy, since it has no speed but has gained height.
We can use the conservation of energy equation:
$E_i = E_f$
where $E_i$ is the initial energy and $E_f$ is the final energy. In this case, we can write:
$\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f$
where $m$ is the mass of the car, $v_i$ is the initial speed (at the bottom of the dip), $h_i$ is the initial height (which is zero), $v_f$ is the final speed (when the car has reached its highest point and its speed is zero), and $h_f$ is the final height (which is what we are trying to find).
We can plug in the given values:
$\frac{1}{2}(500\text{ kg})(16\text{ m/s})^2 + (500\text{ kg})(9.8\text{ m/s}^2)(0\text{ m}) = \frac{1}{2}(500\text{ kg})(0\text{ m/s})^2 + (500\text{ kg})(9.8\text{ m/s}^2)h_f$
Simplifying, we get:
$64000\text{ J} = 4900\text{ J}h_f$
Solving for $h_f$, we get:
$h_f = \frac{64000\text{ J}}{4900\text{ J}} \approx 13.06\text{ m}$
Therefore, the magnitude of the vertical height above the lowest point, when the speed is zero, is approximately 13.06 m.
To find the magnitude of the vertical height above the lowest point when the speed is zero, we can use the principle of conservation of mechanical energy.
The mechanical energy of the roller-coaster car is the sum of its kinetic energy (KE) and potential energy (PE):
ME = KE + PE
At the bottom of the circular dip, all the car's energy is in the form of kinetic energy, since the speed is given as 16 m/s. The potential energy is zero at this point.
ME_bottom = KE_bottom + PE_bottom
ME_bottom = KE_bottom + 0
ME_bottom = KE_bottom
At the point where the speed is zero, all the car's energy is in the form of potential energy. The kinetic energy is zero at this point.
ME_top = KE_top + PE_top
ME_top = 0 + PE_top
ME_top = PE_top
Since mechanical energy is conserved, the mechanical energy at the bottom is equal to the mechanical energy at the top.
ME_bottom = ME_top
KE_bottom = PE_top
The kinetic energy of the roller-coaster car is given by the formula:
KE = (1/2) * m * v^2
where m is the mass of the car and v is the velocity of the car.
Substituting the given values:
KE_bottom = (1/2) * 500 kg * (16 m/s)^2
KE_bottom = 40000 J
Similarly, the potential energy of the roller-coaster car is given by the formula:
PE = m * g * h
where g is the acceleration due to gravity and h is the vertical height.
Substituting the known values:
PE_top = 500 kg * 9.8 m/s^2 * h
Since ME_bottom = ME_top, we can equate the kinetic energy and potential energy:
KE_bottom = PE_top
40000 J = 500 kg * 9.8 m/s^2 * h
Simplifying the equation:
h = 40000 J / (500 kg * 9.8 m/s^2)
h = 8.16 m
Therefore, the magnitude of the vertical height above the lowest point, when the speed is zero, is 8.16 meters.
To find the magnitude of the vertical height above the lowest point when the speed is zero, we can use the conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant if no external forces, such as friction, are acting on it.
At the bottom of the dip, the car has both kinetic energy due to its speed and gravitational potential energy due to its height. At the highest point, the car has zero kinetic energy and maximum potential energy.
The mechanical energy (E) of the roller-coaster car is given by the sum of its kinetic energy (KE) and potential energy (PE):
E = KE + PE
At the bottom of the dip, the car's kinetic energy is given by:
KE = 0.5 * mass * velocity^2
Substituting the given values:
KE = 0.5 * 500 kg * (16 m/s)^2
KE = 0.5 * 500 kg * 256 m^2/s^2
KE = 64,000 Joules
At the highest point, the car's kinetic energy is zero. Therefore, the total mechanical energy (E) is equal to the potential energy at that point:
PE = E
The potential energy (PE) is given by:
PE = mass * gravitational acceleration * height
The gravitational acceleration (g) is approximately 9.8 m/s^2. Substituting the given values:
PE = 500 kg * 9.8 m/s^2 * height
Since the mechanical energy is conserved, we can equate the kinetic energy at the bottom with the potential energy at the highest point:
KE = PE
0.5 * 500 kg * (16 m/s)^2 = 500 kg * 9.8 m/s^2 * height
Solving for the height:
8,000 J = 4,900 kg * m^2/s^2 * height
height = 8,000 J / (4,900 kg * m^2/s^2)
height ≈ 1.633 m
Therefore, the magnitude of the vertical height above the lowest point when the speed is zero is approximately 1.633 meters.