Two motorcycles are traveling due east with different velocities. However, 5.81 seconds later, they have the same velocity. During this 5.81 -second interval, motorcycle A has an average acceleration of 4.07 m/s2 due east, while motorcycle B has an average acceleration of 16.1 m/s2 due east. (a) By how much did the speeds differ at the beginning of the 5.81 -second interval, and (b) which motorcycle was moving faster?
To solve this problem, we can use the kinematic equations of motion. Let's breakdown the problem into two parts: the first part where the motorcycles have different velocities, and the second part where they have the same velocity.
First, let's determine the initial difference in speeds between the motorcycles. Let the initial velocity of motorcycle A be vA, and the initial velocity of motorcycle B be vB. We need to find Δv, the difference in their speeds at the beginning of the 5.81-second interval.
Δv = vB - vA
Now, to find Δv, we'll need to calculate the change in velocity for each motorcycle during the 5.81-second interval. Remember that the change in velocity is equal to the average acceleration multiplied by the time interval.
Change in velocity for motorcycle A:
ΔvA = average acceleration of A * time interval = 4.07 m/s^2 * 5.81 s
Change in velocity for motorcycle B:
ΔvB = average acceleration of B * time interval = 16.1 m/s^2 * 5.81 s
Now, let's substitute these values into the formula for Δv:
Δv = vB - vA
Δv = [vA + ΔvA] - [vB + ΔvB]
Δv = vA + ΔvA - vB - ΔvB
Now, we can rearrange the equation to solve for Δv:
Δv = vA - vB + ΔvA - ΔvB
Substituting the values we know:
Δv = 0 + 4.07 m/s^2 * 5.81 s - 0 - 16.1 m/s^2 * 5.81 s
Simplifying the equation:
Δv = 23.66 m/s - 94.04 m/s
Δv = -70.38 m/s
So, the initial difference in speeds between the motorcycles is -70.38 m/s. Note that the negative sign indicates that motorcycle B was initially faster than motorcycle A.
For part (b) - which motorcycle was moving faster, we can compare the final velocities of the motorcycles.
The final velocity of motorcycle A is:
vA + ΔvA = 0 + 4.07 m/s^2 * 5.81 s
The final velocity of motorcycle B is:
vB + ΔvB = 0 + 16.1 m/s^2 * 5.81 s
Comparing the final velocities, we can see that the magnitude of the change in velocity is the same for both motorcycles. Therefore, both motorcycles have the same final velocity after the 5.81-second interval.
In conclusion, (a) the speeds differ by 70.38 m/s at the beginning of the 5.81-second interval, and (b) motorcycle B was moving faster initially.