Two motorcycles are traveling due east with different velocities. However, four seconds later, they have the same velocity. During this four-second interval, cycle A has an average acceleration of 1.8 m/s2 due east, while cycle B has an average acceleration of 3.8 m/s2 due east. By how much did the speeds differ at the beginning of the four-second interval?
Which motorcycle was moving faster?
To find the initial speed difference between the two motorcycles, we can use the formula:
Acceleration = (Final Velocity - Initial Velocity) / Time
For motorcycle A:
Acceleration = 1.8 m/s^2
Time = 4 seconds
Using the formula, we can rearrange the equation to solve for the initial velocity:
Initial Velocity of A = Final Velocity of A - (Acceleration * Time)
Initial Velocity of A = Final Velocity of A - (1.8 m/s^2 * 4 s)
Similarly, for motorcycle B:
Acceleration = 3.8 m/s^2
Using the formula, we can rearrange the equation to solve for the initial velocity:
Initial Velocity of B = Final Velocity of B - (Acceleration * Time)
Initial Velocity of B = Final Velocity of B - (3.8 m/s^2 * 4 s)
However, we are given that the two motorcycles have the same velocity after 4 seconds. This means their final velocities are equal.
Final Velocity of A = Final Velocity of B
Now we can substitute the final velocities in the above equations:
Initial Velocity of A = Final Velocity of A - (1.8 m/s^2 * 4 s)
Initial Velocity of B = Final Velocity of B - (3.8 m/s^2 * 4 s)
Final Velocity of A = Final Velocity of B
Since the final velocities are equal, we can equate the initial velocities:
Initial Velocity of A - (1.8 m/s^2 * 4 s) = Initial Velocity of B - (3.8 m/s^2 * 4 s)
Simplifying the equation:
Initial Velocity of A - 7.2 m/s = Initial Velocity of B - 15.2 m/s
To find the speed difference at the beginning of the 4-second interval, we subtract the initial velocity of motorcycle B from the initial velocity of motorcycle A:
Initial Speed Difference = (Initial Velocity of A) - (Initial Velocity of B)
Substituting the equation from above:
Initial Speed Difference = (Initial Velocity of A - 7.2 m/s) - (Initial Velocity of B - 15.2 m/s)
Initial Speed Difference = Initial Velocity of A - Initial Velocity of B + 15.2 m/s - 7.2 m/s
Initial Speed Difference = Initial Velocity of A - Initial Velocity of B + 8 m/s
Therefore, the speed difference at the beginning of the 4-second interval is 8 m/s.
To determine which motorcycle was moving faster, compare the initial velocities:
If Initial Velocity of A > Initial Velocity of B, then motorcycle A was moving faster.
If Initial Velocity of A < Initial Velocity of B, then motorcycle B was moving faster.
If Initial Velocity of A = Initial Velocity of B, then they were moving at the same speed.
Check the values for initial velocity to determine this.
To find the difference in speeds between the two motorcycles at the beginning of the four-second interval, we can use the kinematic equation:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Let's assume the initial velocities of cycle A and cycle B are uA and uB respectively. We know that cycle A has an average acceleration of 1.8 m/s^2 and cycle B has an average acceleration of 3.8 m/s^2. We also know that the final velocities for both motorcycles are the same.
Therefore, for cycle A:
vA = uA + (1.8 m/s^2) * (4 s)
And for cycle B:
vB = uB + (3.8 m/s^2) * (4 s)
Since both motorcycles have the same final velocity, vA = vB, we can set the two equations equal to each other:
uA + 1.8 m/s^2 * 4 s = uB + 3.8 m/s^2 * 4 s
Simplifying the equation:
uA + 7.2 m/s = uB + 15.2 m/s
Now, we need to isolate the difference in initial speeds (uB - uA):
uB - uA = (uA + 7.2 m/s) - (uB + 15.2 m/s)
= 7.2 m/s - 15.2 m/s
= -8 m/s
The negative sign indicates that cycle B was initially moving 8 m/s faster than cycle A. However, to determine which motorcycle was moving faster, we need to consider the magnitude of the speed difference. Since the magnitude of -8 m/s is 8 m/s, we can conclude that cycle B was moving faster than cycle A at the beginning of the four-second interval.
If cycle A had velocity a, and
cycle B had velocity b,
a + 1.8*4 = b + 3.8*4
a-b = 8
a was going 8m/s faster than b
makes sense, since b accelerated faster, yet wound up with same speed.