A boat is being pulled by two horses, the first horse at an angle of 40° and the other one at 20°, considering that both the horses are exerting a force of 600N on the boat and the boat weighs 2000kg. Calculate the acceleration and the angle of acceleration

Answer 1

To solve this problem, we need to use vector addition to find the total force exerted on the boat by the two horses. We can then use Newton's second law of motion (F = ma) to calculate the acceleration of the boat.

Step 1: Find the x and y components of the force exerted by each horse.

Let's start by drawing a diagram of the situation:

```
^ Horse 1
/ \
/ \
/ \
-------> Boat
\ /
\ /
\ /
v Horse 2
```

We can see that the force exerted by Horse 1 has two components: one in the x-direction and one in the y-direction. The same is true for Horse 2. We can use trigonometry to calculate these components:

- Horse 1: Fx1 = 600 * cos(40°) = 459.62 N, Fy1 = 600 * sin(40°) = 385.57 N
- Horse 2: Fx2 = 600 * cos(20°) = 563.80 N, Fy2 = 600 * sin(20°) = 205.92 N

Step 2: Find the total force exerted on the boat.

We can add the x and y components of the force from each horse to get the total force exerted on the boat:

- Fx = Fx1 + Fx2 = 459.62 N + 563.80 N = 1023.42 N
- Fy = Fy1 + Fy2 = 385.57 N + 205.92 N = 591.49 N

Now we can use the Pythagorean theorem to find the magnitude of the total force:

- F = sqrt(Fx^2 + Fy^2) = sqrt(1023.42^2 + 591.49^2) = 1176.46 N

We can also use trigonometry to find the angle that the total force makes with the x-axis:

- theta = atan(Fy/Fx) = atan(591.49/1023.42) = 30.70°

Step 3: Calculate the acceleration of the boat.

Now that we know the total force exerted on the boat, we can use Newton's second law of motion to calculate its acceleration:

- F = ma
- a = F/m = 1176.46 N / 2000 kg = 0.588 m/s^2

Step 4: Find the angle of acceleration.

Finally, we can use trigonometry to find the angle of acceleration, which is the angle that the acceleration vector makes with the x-axis:

- alpha = atan(a/Fx) = atan(0.588/1023.42) = 0.033° (rounded to two decimal places)

Therefore, the boat is accelerating at 0.588 m/s^2 and the angle of acceleration is 0.033°.