The motor of a ski boat generates an average power of 7.30 104 W when the boat is moving at a constant speed of 10 m/s. When the boat is pulling a skier at the same speed, the engine must generate an average power of 8.40 104 W. What is the tension in the tow rope that is pulling the skier?
To find the tension in the tow rope, we need to use the equation for power:
Power = Force × Velocity
The power generated by the engine is given (8.40 × 10^4 W), and the velocity of the boat is also given (10 m/s).
When the boat is moving at a constant speed without pulling a skier, the only force acting on the boat is the resistance due to air and water friction. This force is equal to the drag force.
When the boat is pulling a skier at the same speed, there are two forces acting on the boat: the drag force and the tension in the tow rope.
Since the power generated by the engine increases when pulling the skier, it means that the total force acting on the boat has increased.
Let's assume the drag force (F_drag) when the boat is moving without pulling a skier is D.
Therefore, the force exerted by the engine without a skier (F_total1) can be calculated using the power equation:
8.40 × 10^4 W = D × 10 m/s
Similarly, let's assume the tension in the tow rope when the boat is pulling a skier is T.
Therefore, the force exerted by the engine when pulling a skier (F_total2) can be calculated using the power equation:
7.30 × 10^4 W = (D + T) × 10 m/s
Let's solve these equations to find the values of D and T:
First equation:
D = (8.40 × 10^4 W) / (10 m/s)
Second equation:
(D + T) = (7.30 × 10^4 W) / (10 m/s)
Now, substitute the value of D from the first equation into the second equation to solve for T:
T = (7.30 × 10^4 W) / (10 m/s) - (8.40 × 10^4 W) / (10 m/s)
Calculating this, we find T ≈ 4.90 × 10^4 N.
Therefore, the tension in the tow rope that is pulling the skier is approximately 4.90 × 10^4 Newtons.
To find the tension in the tow rope, we can start by using the formula for power:
Power = Force × Velocity
According to the problem, the power generated by the boat when it is moving at a constant speed of 10 m/s is 7.30 × 10^4 W. Therefore, the force exerted by the boat to maintain this speed is:
Force = Power / Velocity
= (7.30 × 10^4 W) / (10 m/s)
= 7300 N
When the boat is pulling a skier at the same speed of 10 m/s, the engine must generate an average power of 8.4 × 10^4 W. This additional power is required to overcome the increased resistance from pulling the skier. Therefore, the additional force exerted by the boat is:
Additional Force = Additional Power / Velocity
= (8.4 × 10^4 W) / (10 m/s)
= 8400 N
The total force exerted by the boat while pulling the skier is:
Total Force = Force + Additional Force
= 7300 N + 8400 N
= 15700 N
Thus, the tension in the tow rope that is pulling the skier is 15700 Newtons.