Question

A car is moving along a road over a hill from point (A) to point (B). At point (A), the hill has a radius (R_1), and the car has speed (v) and tangential acceleration (a_t). At point (B), the hill has a radius (R_2), and the car has constant speed. Which of the following statements about the acceleration of the car at the two points is correct?

a) The tangential acceleration at point (B) is greater than at point (A).

b) The centripetal acceleration at point (A) is greater than at point (B).

c) The tangential acceleration at point (A) is zero.

d) The centripetal acceleration at point (B) is zero.

Answer 1

The correct statement is that the centripetal acceleration at point A is greater than at point B. This is because centripetal acceleration is greater at higher speeds and sharper curves (smaller radii), which are the conditions given for point A.

Step-by-step explanation:

The student is asking about the comparison of centripetal acceleration and tangential acceleration of a car moving over a hill at two different points with different radii of curvature and speeds. Centripetal acceleration (ac) is given by the equation ac = v2/r, where v is the speed of the object and r is the radius of the circular path. At point A, the car is said to have a tangential acceleration and a certain speed while it has no tangential acceleration but a constant speed at point B.

Given this information, we can discern that the tangential acceleration at point B is zero since the car moves at a constant speed (statement a is false). Since the car has tangential acceleration at point A, it cannot be zero (statement c is false). Statement d is also false because centripetal acceleration cannot be zero at point B if the car is still moving along a curved path.

Hence, the correct statement is b), which states that the centripetal acceleration at point A is greater than at point B. This conclusion is based on the fact that centripetal acceleration is proportional to the square of the speed and inversely proportional to the radius. Since point A has a smaller radius (R1) compared to point B (R2) and potentially higher speed (due to acceleration), the centripetal acceleration would be greater at point A.

Answer 2

The correct statement is that the b) centripetal acceleration at point A is greater than at point B because centripetal acceleration is greater with higher speeds and smaller radii. The car at point A has tangential acceleration and potentially a smaller radius than at point B where the car has constant speed.

Step-by-step explanation:

The correct statement about the acceleration of the car at the two points is the following: (b) The centripetal acceleration at point (A) is greater than at point (B). This assertion stems from the nature of centripetal acceleration, which is directly proportional to the square of the velocity (v) and inversely proportional to the radius (R) of the circular path. Given that at point (A), the car has a tangential acceleration (a_t), this implies that the car's speed is increasing, and hence the centripetal acceleration would be higher if the radius is smaller (R_1) compared to point (B) where the car has a constant speed at a potentially larger radius (R_2). Additionally, the tangential acceleration at point (B) cannot be greater than at point (A) as the car is at a constant speed, implying that the tangential acceleration is zero at point (B). Thus, statement (a) is incorrect. Statement (c) is incorrect since at point (A) the tangential acceleration (a_t) is given and not zero. Statement (d) is incorrect since centripetal acceleration is required to keep the car moving in a curve, and at point (B) the car is still moving along a hill, meaning the centripetal acceleration cannot be zero. The centripetal force is what gives the car its tendency to move towards the centre of the circle it's navigating.