A 4.0 x 101 kg wind sled is gliding East across a frozen lake at a constant speed of 12 m/s, when a gust of wind from the southwest exerts a constant force of 1.0 x 102 N on its sails for 3.0 s. With what velocity will the sled be moving after the wind has subsided?
p=mv=40•12 =480 kg•m/s.
F=100 N,
Δt=3 s.
Impulse = F•Δt=100•3=300 kg•m/s.
p is directed to the East
F•Δt=p1 is directed to the northeast.
p and p1 make the angle α=45º
Another angle in the parallelogram of momenta is [360-(2•45)]/2=135º
Use cosine law
p(net) =sqrt(p²+p1²-2p•p1•cos α) =
=sqrt(480²+ 300² -2•480•300•cos 135º)=
=sqrt(480²+ 300² +2•480•300•0.707)= 723.9 kg•m/s.
v=p(net)/m =723.9/40=18.1 m/s
To calculate the final velocity of the sled after the wind has subsided, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
1. First, let's determine the acceleration of the sled caused by the gust of wind. We can use the equation:
F = m * a
Where F is the force exerted by the wind, m is the mass of the sled, and a is the acceleration.
2. Rearranging the equation, we can solve for acceleration (a):
a = F / m
Plugging in the values:
a = (1.0 x 10^2 N) / (4.0 x 10^1 kg)
a = 2.5 m/s^2
Therefore, the acceleration of the sled caused by the gust of wind is 2.5 m/s^2.
3. Since the initial velocity of the sled is given as 12 m/s in the eastward direction and the force of the gust of wind is acting towards the southwest, we need to break down the vectors to consider their components.
The southwest wind force can be resolved into two components: one acting in the eastward direction and another acting in the northward direction.
The magnitude of the eastward component of the wind force can be found using trigonometry:
Eastward component = Force * cos(45 degrees)
Plugging in the values:
Eastward component = (1.0 x 10^2 N) * cos(45 degrees)
Eastward component = 70.7 N
4. Since there are no other forces acting on the sled, the net force acting on the sled is equal to the eastward component of the wind force:
Net force = Eastward component
Net force = 70.7 N
5. Now using the formula, we can calculate the final velocity of the sled:
v = u + at
Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Plugging in the values:
v = 12 m/s + (70.7 N) / (4.0 x 10^1 kg) * 3.0 s
v = 12 m/s + 5.3 m/s
v = 17.3 m/s
Therefore, after the wind has subsided, the sled will be moving at a velocity of 17.3 m/s in the eastward direction.