At t = 0, an automobile traveling north begins to make a turn. It follows one-quarter of the arc of a circle of radius 9.3 m until, at t = 1.38 s, it is traveling east. The car does not alter its speed during the turn.
(a) Find the car's speed.
___ m/s
(b) Find the change in its velocity during the turn.
____ m/s
(c) Find its average acceleration during the turn.
To solve this problem, we need to use the concepts of kinematics and circular motion. Let's break down the problem into steps:
Step 1: Find the car's displacement during the turn.
We know that the car follows one-quarter of the arc of a circle. To find the displacement, we need to calculate the length of this arc. The formula for the length of an arc in a circle is given by:
Arc length (s) = circumference (C) * (angle (θ) / 360)
In this case, the angle is 90 degrees (one-quarter of a circle). The circumference can be calculated using the formula C = 2πr, where r is the radius. Given that the radius is 9.3 m, we can substitute the values into the formula to find the arc length.
Step 2: Find the car's velocity.
Velocity (v) is defined as the rate of change of displacement. Since we know the displacement and time, we can calculate the car's velocity using the formula v = displacement / time.
Step 3: Find the change in velocity.
The change in velocity is the difference between the final velocity and the initial velocity. In this case, the car initially travels north and ends up traveling east. We can use vector addition to find the change in velocity.
Step 4: Find the average acceleration.
Acceleration (a) is defined as the rate of change of velocity. The average acceleration during the turn can be calculated using the formula a = change in velocity / time.
Now let's solve the problem:
Step 1: Find the car's displacement during the turn.
Arc length (s) = (2π(9.3) * 90) / 360
= 14.645 m
Step 2: Find the car's velocity.
Velocity (v) = displacement / time
= 14.645 m / 1.38 s
≈ 10.620 m/s
Step 3: Find the change in velocity.
Since the car starts by traveling north and ends up traveling east, the change in velocity is the vector sum of these two velocities. We can use the Pythagorean theorem to calculate the magnitude of the change in velocity.
Change in velocity (Δv) = square root of (north velocity^2 + east velocity^2)
= square root of (10.62^2 + 0^2)
≈ 10.62 m/s
Step 4: Find the average acceleration.
Average acceleration (a) = change in velocity / time
= 10.62 m/s / 1.38 s
≈ 7.696 m/s^2
Therefore, the answers to the questions are:
(a) The car's speed is approximately 10.620 m/s.
(b) The change in velocity during the turn is approximately 10.620 m/s.
(c) The average acceleration during the turn is approximately 7.696 m/s^2.