At t = 0, an automobile traveling north begins to make a turn. It follows one-quarter of the arc of a circle of radius 10.1 m until, at t = 1.64 s, it is traveling east. The car does not alter its speed during the turn.
(a) Find the car's speed.
(b) Find the change in its velocity during the turn.
(c) Find its average acceleration during the turn.
To solve this problem, we need to analyze the motion of the car during the turn.
(a) To find the car's speed, we need to use the formula for the circumference of a circle. At t = 0, the car starts at one point on the circle and ends at another point 1/4th of the circumference away. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle.
Here, the radius is given as 10.1 m. So, the total distance traveled by the car during the turn is 1/4th of the circumference, which is (1/4) * 2π * 10.1 = 5.05π m.
The time interval taken for the turn is given as t = 1.64 s. The car's speed can be calculated by dividing the distance by the time taken:
Speed = Distance / Time = (5.05π) / 1.64 ≈ 4.88 m/s (rounded to two decimal places).
Therefore, the car's speed is approximately 4.88 m/s.
(b) To find the change in velocity during the turn, we need to consider the direction of motion. At t = 0, the car is traveling north, and at t = 1.64 s, it is traveling east. The change in velocity is the difference between the initial and final velocities, taking into account their respective directions.
Since the car's speed remains constant throughout the turn, the magnitude of the initial velocity and final velocity is the same. We can find the magnitude of the velocity using the formula:
Speed = Magnitude of Velocity = √(Vx^2 + Vy^2),
where Vx and Vy represent the x and y components of velocity.
At t = 0, the car is traveling north with all its velocity along the y-axis. So, Vy = Speed, and Vx = 0.
At t = 1.64 s, the car is traveling east with all its velocity along the x-axis. So, Vx = Speed, and Vy = 0.
Since the magnitude of velocity is the same in both cases, we have:
Speed = √(Vx^2 + Vy^2) = √(0^2 + Speed^2) = √(Speed^2) = Speed.
Therefore, the change in velocity during the turn is 0 m/s.
(c) To find the average acceleration during the turn, we can use the formula:
Average Acceleration = Change in Velocity / Time Interval.
Since the change in velocity is 0 m/s (as calculated in part (b)), and the time interval is 1.64 s, the average acceleration is:
Average Acceleration = 0 m/s / 1.64 s = 0 m/s².
Therefore, the average acceleration during the turn is 0 m/s².