A 2-kg object has a velocity of 22i m/s at t = 0. A constant resultant force of (2.0i + 4.0j) N then acts on the object for 12 s. What is the magnitude of the object’s velocity at the end of the interval?
To find the final velocity of the object, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Given that the object has a mass of 2 kg and a constant resultant force of (2.0i + 4.0j) N acting on it, we can calculate the acceleration of the object.
First, we need to find the net force acting on the object. Since the force is constant, we can calculate the net force by taking the sum of the individual force components:
Net force = (2.0i + 4.0j) N
Next, we can use Newton's second law to calculate the acceleration:
Net force = mass x acceleration
(2.0i + 4.0j) N = 2 kg x acceleration
Simplifying, we have:
2.0i + 4.0j = 2 kg x acceleration
Comparing the components, we have:
2.0 = 2 x acceleration in the x-direction (i)
4.0 = 2 x acceleration in the y-direction (j)
Solving for acceleration in each direction:
Acceleration in the x-direction (i) = 1.0 m/s²
Acceleration in the y-direction (j) = 2.0 m/s²
Now, we have the acceleration of the object. We can use this to find the change in velocity of the object during the 12-second interval.
To do this, we can use the kinematic equation:
Δv = a x Δt
Δv is the change in velocity,
a is the acceleration, and
Δt is the time interval.
In this case, the acceleration in the x-direction is 1.0 m/s², and the time interval is 12 seconds. Therefore:
Δv = (1.0 m/s²) x (12 s)
Δv = 12 m/s
So, the object's velocity changes by 12 m/s during the 12-second interval.
Finally, we need to find the final velocity of the object. Since the initial velocity of the object is given as 22i m/s, we can find the final velocity by adding the change in velocity to the initial velocity:
Final velocity = Initial velocity + Δv
Final velocity = 22i m/s + 12 m/s
Final velocity = 34i m/s
Therefore, at the end of the interval, the magnitude of the object's velocity is 34 m/s.
To find the magnitude of the object's velocity at the end of the interval, we need to calculate the change in velocity caused by the constant resultant force over the given time interval.
The change in velocity can be determined using Newton's second law of motion:
F = ma
Where F is the force, m is the mass of the object, and a is the acceleration experienced by the object.
First, let's calculate the acceleration:
To find the acceleration, we divide the force by the mass of the object:
a = F/m
Given:
Force, F = (2.0i + 4.0j) N
Mass, m = 2 kg
Substitute the values:
a = (2.0i + 4.0j) N / 2 kg
Simplifying,
a = 1.0i + 2.0j m/s^2
Next, we can calculate the change in velocity by multiplying the acceleration by the time interval:
Δv = a * t
Given:
Time interval, t = 12 s
Acceleration, a = 1.0i + 2.0j m/s^2
Substitute the values:
Δv = (1.0i + 2.0j) m/s^2 * 12 s
Simplifying,
Δv = 12i + 24j m/s
Finally, to find the magnitude of the object's velocity at the end of the interval, we need to add the initial velocity to the change in velocity:
v = vo + Δv
Given:
Initial velocity, vo = 22i m/s
Change in velocity, Δv = 12i + 24j m/s
Substitute the values:
v = (22i m/s) + (12i + 24j) m/s
Simplifying,
v = 34i + 24j m/s
The magnitude of the object's velocity at the end of the interval can be calculated using the Pythagorean theorem:
|v| = sqrt(vx^2 + vy^2)
Given:
vx = 34 m/s (velocity component along the x-axis)
vy = 24 m/s (velocity component along the y-axis)
Substitute the values:
|v| = sqrt((34 m/s)^2 + (24 m/s)^2)
Simplifying,
|v| = sqrt(1156 m^2/s^2 + 576 m^2/s^2)
|v| = sqrt(1696 m^2/s^2)
|v| = 41.2 m/s
Therefore, the magnitude of the object's velocity at the end of the interval is 41.2 m/s
v = 22i + at
a = F/m
= 22i + (2i+4j)/2 * 12
= 22i + 12i+24j
= 34i+24j
|v| = √(34^2 + 24^2)