Calculate the change in momentum
The initial momentum \(p_{initial}\) is given by \(m \cdot v_{initial}\), where \(v_{initial}\) is the initial velocity (northward). The final momentum \(p_{final}\) is given by \(m \cdot v_{final}\).
Calculating the initial and final momenta:
\(p_{initial}= 3.0\, kg \times 15\, m/s = 45\, kg\cdot m/s^{(north)}\)
\(p_{final}= 3.0\, kg \times 25\, m/s = 75\, kg\cdot m/s\)
Now we can find the change in momentum \(\Delta p = p_{final} - p_{initial}\). Since the initial and final velocities have different directions, the change in momentum also has a direction. We will calculate the change in momentum in the horizontal and vertical directions separately.
Change in horizontal momentum:
\(\Delta p_x = 3.0\, kg \cdot \left( 20\, m/s\right) - 0 = 60\, kg\cdot m/s^{(east)}\)
Change in vertical momentum:
\(\Delta p_y = 3.0\, kg \cdot \left( 15\, m/s\right) - 3.0\, kg \cdot \left( 15\, m/s\right) = 0\, kg\cdot m/s\)
To find the magnitude of the change in momentum, use the Pythagorean theorem:
\(\Delta p = \sqrt{(\Delta p_x)^2 + (\Delta p_y)^2}\)
Calculating the change in momentum:
\(\Delta p = \sqrt{(60\, kg\cdot m/s)^2 + (0\, kg\cdot m/s)^2} = 60\, kg\cdot m/s\)
(a) The final velocity of the body after 4.0 seconds is \(25 \,m/s\).
(b) The change in momentum during the 4.0-second interval is \(60\, kg\cdot m/s\).
