A car starts from rest and travels along a circular track with a radius of 55.0 meters. The car starts out at a position due East of the center of the track, heading due North initially around the track, which is counterclockwise. The car increases its speed by 105 m/s every minute around the track. What is the cars acceleration towards the center of the track when it is 1/4 of the way around the track?
To find the car's acceleration towards the center of the track when it is 1/4 of the way around the track, we can use the centripetal acceleration formula.
The centripetal acceleration (a) of an object moving in a circular path is given by the formula:
a = v^2 / r
Where:
a = centripetal acceleration
v = velocity of the car
r = radius of the circular track
First, we need to find the car's velocity (v) when it is 1/4 of the way around the track.
The car increases its speed by 105 m/s every minute around the track. Since the car is initially heading due North, the velocity component in the north direction increases at a rate of 105 m/s per minute.
At 1/4 of the way around the track, the car has traveled 1/4 * 2πr = 1/4 * 2π * 55.0 meters = 34.54 meters.
To find the car's velocity at this position, we use the following equation:
v = v0 + at
Where:
v = velocity at the current position
v0 = initial velocity (at the starting point)
a = acceleration (change in velocity per unit time)
t = time
Since the car is starting from rest (v0 = 0), and the acceleration is 105 m/s per minute, we can substitute these values into the equation:
v = 0 + (105 m/s/min) * (34.54 m / 105 m/s)
Simplifying:
v ≈ 34.54 m/s
Now, we have the velocity (v) and the radius (r). We can substitute these values into the centripetal acceleration formula to find the car's acceleration towards the center of the track when it is 1/4 of the way around the track:
a = (v^2) / r
= (34.54 m/s)^2 / 55.0 m
Calculating:
a ≈ 21.67 m/s^2
Therefore, the car's acceleration towards the center of the track when it is 1/4 of the way around the track is approximately 21.67 m/s^2.
To find the car's acceleration towards the center of the track when it is 1/4 of the way around the track, we need to use the concept of centripetal acceleration.
Centripetal acceleration (a) is the acceleration of an object moving in a circular path at a constant speed and is given by the formula:
a = v^2 / r
where v is the velocity of the object and r is the radius of the circular path.
In this case, the car is increasing its speed by 105 m/s every minute around the track. Let's calculate the velocity of the car when it is 1/4 of the way around the track.
1. Determine the distance the car has traveled when it is 1/4 of the way around the track.
Distance = (1/4) * (circumference of the track)
= (1/4) * (2π * radius)
= (1/4) * (2π * 55.0 m)
2. To find the time it takes to travel this distance, we need to know the car's speed.
The car increases its speed by 105 m/s every minute around the track.
So, after t minutes, the car's speed would be 105 * t m/s.
3. Set up an equation where the distance traveled equals the speed multiplied by the time.
(1/4) * (2π * 55.0 m) = (105 * t) m/s * t
4. Solve for t by rearranging the equation:
(2π * 55.0 m) / 4 = 105 * t^2 m^2/s^2
5. Simplify the equation:
(2π * 55.0 m) / (4 * 105) = t^2
t^2 = (2π * 55.0 m) / (4 * 105)
t = √[(2π * 55.0 m) / (4 * 105)]
Now that we have the value of t, let's calculate the velocity at t minutes.
6. Velocity = 105 * t m/s
Finally, substitute the values of velocity (v = 105 * t m/s) and radius (r = 55.0 m) into the centripetal acceleration formula.
7. Acceleration = v^2 / r
= (105 * t)^2 / 55.0 m
Evaluate this expression to find the acceleration towards the center of the track when the car is 1/4 of the way around the track.