A 3.0-kg body is initially moving northward at 15 m/s. Then a force of 15 N,
toward the east, acts on it for 4.0 s. (a) At the end of the 4.0 s, what is the body’s final velocity?
(b) What is the change in momentum during the 4.0 s?
To find the final velocity of the body and the change in momentum during the 4.0 seconds, we can use the principles of Newton's second law of motion and the equation for momentum.
(a) To find the final velocity, we need to calculate the acceleration produced by the force and then use it to find the change in velocity.
Step 1: Calculate acceleration using Newton's second law of motion:
The force acting on the body is 15 N.
The mass of the body is 3.0 kg.
Using the formula F = ma, we can solve for acceleration:
a = F / m
Substituting the values, we get:
a = 15 N / 3.0 kg
a = 5 m/s^2
Step 2: Calculate the change in velocity:
Given that the force acts for 4.0 seconds, we can use the equation of motion:
Δv = a * t
Substituting the values, we get:
Δv = 5 m/s^2 * 4.0 s
Δv = 20 m/s
Step 3: Add the change in velocity to the initial velocity to get the final velocity:
The initial velocity of the body is 15 m/s (northward), and the change in velocity is 20 m/s (eastward). Remember that velocity has both magnitude and direction.
Using vector addition, we can calculate the final velocity using the Pythagorean theorem:
v^2 = (15 m/s)^2 + (20 m/s)^2
v^2 = 225 m^2/s^2 + 400 m^2/s^2
v^2 = 625 m^2/s^2
Taking the square root of both sides:
v = √(625 m^2/s^2)
v = 25 m/s
Therefore, the body's final velocity is 25 m/s to the northeast.
(b) To find the change in momentum, we use the equation:
Δp = mvf - mvi
Given:
Mass (m) = 3.0 kg
Initial velocity (vi) = 15 m/s (northward)
Final velocity (vf) = 25 m/s (northeast)
Substituting the values, we get:
Δp = (3.0 kg)(25 m/s) - (3.0 kg)(15 m/s)
Δp = 75 kg·m/s - 45 kg·m/s
Δp = 30 kg·m/s
Therefore, the change in momentum during the 4.0 seconds is 30 kg·m/s to the northeast.