Car A is moving at a constant speed Va=75 km/h, while car C is moving at a constant speed Vc=42 km/h on a circular exit ramp with radius p=80m. Determine the velocity and acceleration of C relative to A.
To determine the velocity and acceleration of car C relative to car A, we need to consider the relative motion between the two cars.
Velocity of C relative to A:
The velocity of car C relative to car A can be calculated by subtracting the velocity of car A from the velocity of car C. Since the two cars are moving in a circular motion, we can use the concept of relative velocities.
The velocity of car C relative to car A can be determined using the formula:
Vc/a = Vc - Va
Given:
Va = 75 km/h
Vc = 42 km/h
Converting the velocities to m/s:
Va = 75 km/h * (1000 m/km) / (3600 s/h) = 20.83 m/s
Vc = 42 km/h * (1000 m/km) / (3600 s/h) = 11.67 m/s
Therefore, the velocity of car C relative to car A is:
Vc/a = 11.67 m/s - 20.83 m/s = -9.16 m/s (negative sign indicates opposite direction)
Acceleration of C relative to A:
The acceleration of car C relative to car A can be determined by subtracting the acceleration of car A from the acceleration of car C. However, as the problem statement does not provide any information about the accelerations, we cannot calculate the acceleration of car C relative to car A without additional information.
Therefore, the velocity of car C relative to car A is -9.16 m/s, but we cannot determine the acceleration of car C relative to car A without more information.
To determine the velocity and acceleration of Car C relative to Car A, we need to analyze their motion on the circular exit ramp.
Velocity:
The velocity of an object moving in a circle is given by the equation v = rω, where v is the linear velocity, r is the radius of the circular path, and ω (omega) is the angular velocity.
1. For Car A:
Given the constant speed Va of Car A is 75 km/h, we need to convert it to meters per second (m/s). Since 1 km = 1000 m and 1 hour = 3600 seconds, we have:
Va = 75 km/h = (75 * 1000) m / (3600 s) = 20.83 m/s.
2. For Car C:
The linear velocity of Car C on the circular exit ramp is the same as its constant speed Vc, as Car C is moving tangentially to the circular path. Given Vc = 42 km/h, we convert it to m/s using the same conversion factor:
Vc = 42 km/h = (42 * 1000) m / (3600 s) = 11.67 m/s.
The velocity of Car C relative to Car A is the difference between their velocities:
Vc_A = Vc - Va = 11.67 m/s - 20.83 m/s = -9.16 m/s (negative because their velocities are in opposite directions).
Acceleration:
The acceleration of an object moving in a circle is given by the equation a = rω², where a is the centripetal acceleration, r is the radius of the circular path, and ω (omega) is the angular velocity.
1. For Car A:
Since Car A is moving at a constant speed on a circular path, it experiences no net acceleration. Therefore, its centripetal acceleration is zero: aA = 0 m/s².
2. For Car C:
Car C is moving on a circular path. The radius of the circular exit ramp, p = 80 m, is given, and we can derive the angular velocity ω of Car C using the equation v = rω. Rearranging the equation, we have ω = v / r.
ω = Vc / p = 11.67 m/s / 80 m = 0.146 s⁻¹.
Now, we can calculate the centripetal acceleration of Car C:
aC = pω² = (80 m) * (0.146 s⁻¹)² = 1.06 m/s².
The acceleration of Car C relative to Car A is the difference between their accelerations:
aC_A = aC - aA = 1.06 m/s² - 0 m/s² = 1.06 m/s².