A 6.90 kg penguin runs onto a huge sheet of frictionless Arctic ice. At t=0 it is at x=0 and y=0 with an initial velocity of 0.47 m/s along the positive x-axis. It slides while being pushed by the wind with a force of 0.49 N directed along the positive y-axis. Calculate the magnitude of the penguin's velocity at t = 8.69 s.
To solve this problem, we can break down the forces acting on the penguin and use Newton's second law of motion (F = ma) to find its acceleration. We can then integrate the acceleration with respect to time to find the penguin's velocity at any given time. Here's how we can approach this:
1. Determine the net force acting on the penguin:
The penguin is being pushed by the wind with a force of 0.49 N along the positive y-axis. Since there are no other forces mentioned in the problem statement, the net force acting on the penguin is equal to the force from the wind, which is 0.49 N.
2. Find the acceleration of the penguin:
According to Newton's second law, the net force on an object is equal to its mass multiplied by its acceleration (F = ma). In this case, the mass of the penguin is given as 6.90 kg. Therefore, the acceleration can be calculated as follows:
a = F / m
a = 0.49 N / 6.90 kg
a ≈ 0.071 m/s^2
3. Integrate the acceleration to find the velocity:
Since we know the initial velocity (0.47 m/s along the positive x-axis) and the time (8.69 s), we can determine the penguin's velocity at t = 8.69 s.
The velocity of the penguin can be calculated as the area under the acceleration-time graph. Since the acceleration is constant, we can use the formula:
v = u + at
where:
v: Final velocity
u: Initial velocity
a: Acceleration
t: Time
Plugging in the values, we get:
v = 0.47 m/s + (0.071 m/s^2) * (8.69 s)
v ≈ 0.47 m/s + 0.61799 m/s
v ≈ 1.088 m/s
Therefore, the magnitude of the penguin's velocity at t = 8.69 s is approximately 1.088 m/s.