A race car starts from rest on a circular track. The car increases its speed at a constant rate at as it goes 2.75 times around the track. Find the angle that the total acceleration of the car makes with the radius connecting the center of the track and the car at the moment the car completes its trip of 2.75 times around the circle.
To find the angle that the total acceleration of the car makes with the radius connecting the center of the track and the car, we need to break down the problem into smaller steps:
Step 1: Understand the problem
A race car starts from rest on a circular track and increases its speed at a constant rate. We need to find the angle that the total acceleration makes with the radius at the moment the car completes its trip of 2.75 times around the circle.
Step 2: Define the variables
Let's denote the total acceleration as "a" and the radius connecting the center of the track and the car as "r". We are trying to find the angle "θ" between the total acceleration and the radius.
Step 3: Determine the speed of the car
Since the car is increasing its speed at a constant rate, we can use the kinematic equation to find the final speed of the car. The equation is:
v = u + at
Where:
v is the final velocity,
u is the initial velocity (which is 0 in this case),
a is the acceleration, and
t is the time taken.
Step 4: Calculate the time taken to complete the trip
Since the car completes 2.75 times around the track, we can use the formula for the circumference of a circle to find the distance traveled:
circumference = 2πr
The distance traveled by the car is 2.75 times the circumference, so:
distance = 2.75 * circumference
We can also use the formula for speed:
speed = distance / time
Since the initial velocity is 0, the final velocity is the speed of the car. We can rearrange the equation to solve for time:
time = distance / speed
Step 5: Calculate the total acceleration
The total acceleration can be calculated using the formula:
a = (v - u) / t
Since the initial velocity is 0, this simplifies to:
a = v / t
Step 6: Calculate the angle θ
To find the angle between the total acceleration and the radius, we can use the formula:
θ = arctan(a / g)
Where g is the acceleration due to gravity.
Step 7: Plug in the values and solve
Let's suppose the radius is 10 meters, and the constant rate of acceleration is 5 m/s².
1. Calculate the speed of the car using the kinematic equation:
v = u + at
v = 0 + (5 m/s²) * t
v = 5t m/s
2. Calculate the time taken to complete the trip:
distance = 2.75 * (2π * 10)
distance = 2.75 * 20π
distance ≈ 172.788 meters
speed = distance / time
5t = 172.788 / t
5t² = 172.788
t² = 34.5576
t ≈ 5.873 seconds
3. Calculate the total acceleration:
a = v / t
a = (5t) / t
a = 5 m/s²
4. Calculate the angle θ using the arctan formula:
θ = arctan(a / g)
θ = arctan(5 m/s² / 9.8 m/s²)
θ ≈ 28.094 degrees
Therefore, the angle that the total acceleration of the car makes with the radius connecting the center of the track and the car at the moment the car completes its trip of 2.75 times around the circle is approximately 28.094 degrees.