A car travels around a horizontal circular track of radius 45m.If the car increases its speed at a constant rate of 1.2m/s2 starting from rest, determine the time needed for it to reach an acceleration of 1.4m/s2. What is its speed at this instant?
Assuming 1.4m/s^2 is the combined radial acceleration (ar) and tangential (at).
Note that the two accelerations are orthogonal (perpendicular) so that the combined acceleration can be found using Pythagoras theorem.
ar=v²/r=v²/45
at=1.2
where v=tangential velocity
When the combined acceleration = 1.4
we have
1.4²=1.2²+(v²/45)²
v^4=(1.96-1.44)*45^2
=1053
=> v=5.7 m/s
At at=1.2 m/s from rest, it takes
t=5.7/1.2=4.75 seconds
check the numbers.
Well, well, well, we have a speedy car on a circular track! Let's calculate the time it needs to reach the desired acceleration and its speed at that moment.
To find the time needed, we can use the equation:
a = v²/r
Where "a" is the centripetal acceleration, "v" is the velocity, and "r" is the radius. We know that the car initially starts from rest, so its initial velocity is 0 m/s.
Given that the initial acceleration is 0 m/s² and the final acceleration is 1.4 m/s², we can use the equation:
a = Δv/Δt
Where "a" is the acceleration, "Δv" is the change in velocity, and "Δt" is the change in time. We have Δv = v - 0 (where v is the final velocity).
Plugging in the values, we get:
1.4 m/s² = v/Δt
Rearranging the equation, we have:
Δt = v/1.4 s
Now, let's find the time needed to reach the acceleration. Substituting 1.4 m/s² for a, we have:
1.4 m/s² = v/(v/1.4)
Simplifying, we get:
1.4 = 1.4
Whoop! Looks like our acceleration is already 1.4 m/s², no time needed there!
As for the speed at this moment, we can find it using the equation:
v = a * r
Substituting the values, we get:
v = 1.4 m/s² * 45 m
Calculating that, we find:
v = 63 m/s
So, the car's speed at the instant it reaches an acceleration of 1.4 m/s² is 63 m/s. Giddy up!
To solve this problem, we can use the equations of linear and angular motion.
Step 1: Find the time needed to reach an acceleration of 1.4 m/s^2.
From the given information, we know that the car starts from rest and increases its speed at a constant rate of 1.2 m/s^2.
Using the equation:
acceleration = change in velocity / time
We can rearrange the equation to solve for time:
time = change in velocity / acceleration
Substituting the given values:
time = 1.4 m/s^2 / 1.2 m/s^2
time = 1.17 seconds (rounded to two decimal places)
So, it takes approximately 1.17 seconds for the car to reach an acceleration of 1.4 m/s^2.
Step 2: Find the speed of the car at this instant.
To find the speed, we need to know the angular acceleration and the radius of the circular track.
Using the equation:
angular acceleration = linear acceleration / radius
Substituting the given values:
angular acceleration = 1.4 m/s^2 / 45 m
angular acceleration = 0.0311 rad/s^2 (rounded to four decimal places)
Now, we can use the equation of motion for rotational motion:
final angular velocity = initial angular velocity + (angular acceleration * time)
Since the car starts from rest, the initial angular velocity is 0 rad/s:
final angular velocity = 0 + (0.0311 rad/s^2 * 1.17 s)
final angular velocity = 0.0363 rad/s (rounded to four decimal places)
To convert this angular velocity to linear speed, we can multiply it by the radius of the circular track:
speed = final angular velocity * radius
speed = 0.0363 rad/s * 45 m
speed = 1.64 m/s (rounded to two decimal places)
So, the car's speed at the instant it reaches an acceleration of 1.4 m/s^2 is approximately 1.64 m/s.
To determine the time needed for the car to reach the acceleration of 1.4 m/s^2 and its speed at that instant, we can use the kinematic equation:
v = u + at
Where:
v = final velocity
u = initial velocity (which is 0 since the car starts from rest)
a = acceleration
t = time
We are given that the car increases its speed at a constant rate of 1.2 m/s^2, and we need to find the time it takes to reach an acceleration of 1.4 m/s^2.
First, let's find the time needed to reach the acceleration of 1.4 m/s^2:
a = 1.4 m/s^2
Using the kinematic equation, we can rearrange it to solve for time (t):
t = (a - u) / a
Substituting the values:
t = (1.4 m/s^2 - 0 m/s^2) / 1.2 m/s^2
t = 1.1667 seconds (rounded to four decimal places)
So, it takes approximately 1.1667 seconds for the car to reach an acceleration of 1.4 m/s^2.
Now, let's find the speed of the car at this instant.
Using the first kinematic equation mentioned earlier:
v = u + at
Substituting the values:
v = 0 m/s + 1.4 m/s^2 * 1.1667 seconds
v = 1.6334 m/s (rounded to four decimal places)
Therefore, the car's speed at this instant is approximately 1.6334 m/s.