A race car starts from rest on a circular track of radius 400 m. Its speed increases at the constant rate of 0.500 m/s2. At the point where the magnitudes of the radial acceleration is twice the tangential acceleration, determine (a) the speed of the race car, and (b) the elapsed time.
hey wants the answer?
tangential a = 0.5 m/s^2
so
v = 0.5 t
radial a = v^2/R = 0.25 t^2/ 400 = 0.000625 t^2
at radial acc = 1.0 m/s^2 which is twice a
1.0 = 0.000625 t^2
t^2 = 1600
t = 40 seconds
v = 0.5 t = 20 m/s
big hurry?
To answer this question, we need to analyze the radial and tangential acceleration of the race car and then use them to find the speed and elapsed time.
Let's start by understanding the radial and tangential acceleration:
1. Radial Acceleration (ar):
Radial acceleration is the acceleration that points towards the center of the circular path. It is given by the formula: ar = v² / r, where v is the velocity (speed) and r is the radius of the circular track.
2. Tangential Acceleration (at):
Tangential acceleration is the acceleration along the tangent to the circular path. It is given by the formula: at = dv / dt, where dv is the change in velocity and dt is the change in time.
In this problem, we are given that the magnitude of the radial acceleration is twice the tangential acceleration. Mathematically, we can represent this as:
(ar) = 2(at)
Now, let's proceed to find the speed of the race car:
Step 1: Calculate the tangential acceleration (at) using the given information:
Given: ar = 2(at)
Using the formula for radial acceleration (ar = v² / r), we can rewrite this equation as:
v² / r = 2(at)
Step 2: Rearrange the equation to solve for tangential acceleration (at):
at = v² / (2r)
Step 3: Substitute the value of tangential acceleration in the equation to find the speed:
at = 0.500 m/s² (given)
Substituting this value, we get:
0.500 m/s² = v² / (2 * 400 m)
0.500 m/s² = v² / 800 m
v² = 0.500 m/s² * 800 m
v² = 400 m²/s²
v = sqrt(400 m²/s²)
v = 20 m/s
Therefore, the speed of the race car is 20 m/s.
Next, let's find the elapsed time:
Step 1: Use the given information to calculate the tangential acceleration (at):
Given: at = 0.500 m/s²
Step 2: Use the equation for tangential acceleration (at = dv / dt) to find the change in velocity:
0.500 m/s² = dv / dt
dv = 0.500 m/s² * dt
Step 3: Integrate both sides of the equation to find the change in velocity (dv):
∫dv = ∫0.500 m/s² * dt
v - 0 m/s = 0.500 m/s² * t
Step 4: Rearrange the equation to solve for the elapsed time (t):
t = (v - 0 m/s) / 0.500 m/s²
t = v / 0.500 m/s²
t = 20 m/s / 0.500 m/s²
t = 40 s
Therefore, the elapsed time is 40 seconds.
In conclusion:
(a) The speed of the race car is 20 m/s.
(b) The elapsed time is 40 seconds.