A person in a kayak starts paddling, and it accelerates from 0 to 0.8 miles/hour in a distance of 0.8 km. If the combined mass of the person and the kayak is 80 kg, what is the magnitude of the net force acting on the kayak
The Correct Answer and Explanation is:
To solve this problem, we need to apply Newton’s second law of motion and kinematic equations.
Step 1: Convert units
We are given that the initial velocity v0=0v_0 = 0v0=0 and the final velocity vf=0.8v_f = 0.8vf=0.8 miles/hour. We need to convert this to meters per second (m/s) for consistency with the SI units.1 mile=1609.34 meters,1 hour=3600 seconds1 \text{ mile} = 1609.34 \text{ meters}, \quad 1 \text{ hour} = 3600 \text{ seconds}1 mile=1609.34 meters,1 hour=3600 secondsvf=0.8 miles/hour=0.8×1609.343600≈0.357 m/sv_f = 0.8 \text{ miles/hour} = \frac{0.8 \times 1609.34}{3600} \approx 0.357 \text{ m/s}vf=0.8 miles/hour=36000.8×1609.34≈0.357 m/s
Step 2: Find acceleration
Now, we can use the kinematic equation to find the acceleration aaa, since we are given the initial and final velocities and the distance traveled.vf2=v02+2adv_f^2 = v_0^2 + 2advf2=v02+2ad
Where:
- vf=0.357v_f = 0.357vf=0.357 m/s (final velocity)
- v0=0v_0 = 0v0=0 m/s (initial velocity)
- d=0.8d = 0.8d=0.8 km = 800 m (distance)
Substitute the values into the equation:(0.357)2=0+2a(800)(0.357)^2 = 0 + 2a(800)(0.357)2=0+2a(800)0.127≈1600a0.127 \approx 1600a0.127≈1600aa≈0.1271600≈0.0000794 m/s2a \approx \frac{0.127}{1600} \approx 0.0000794 \text{ m/s}^2a≈16000.127≈0.0000794 m/s2
Step 3: Apply Newton’s Second Law
Now, we apply Newton’s second law to find the net force FFF, which is given by:F=maF = maF=ma
Where:
- m=80m = 80m=80 kg (combined mass of the person and the kayak)
- a=0.0000794a = 0.0000794a=0.0000794 m/s² (acceleration)
Substitute the values:F=80×0.0000794≈0.00635 NF = 80 \times 0.0000794 \approx 0.00635 \text{ N}F=80×0.0000794≈0.00635 N
Final Answer:
The magnitude of the net force acting on the kayak is approximately 0.0064 N.
Explanation:
The net force required to accelerate the kayak to 0.8 miles/hour over 0.8 km is very small due to the low acceleration. This is because the final velocity is relatively small and the distance over which the acceleration occurs is not very large either.
