A 50.0 kg object is moving at 18.2 m/s when a 200 N force
is applied opposite the direction of the object’s motion,
causing it to slow down to 12.6 m/s.
How long was this force applied?
The Correct Answer and Explanation is :
To determine how long the force was applied, we can use the concepts of kinematics and Newton’s second law of motion. Here’s a step-by-step breakdown of the calculation and explanation:
Given Data
- Mass of the object, ( m = 50.0 \, \text{kg} )
- Initial velocity, ( v_i = 18.2 \, \text{m/s} )
- Final velocity, ( v_f = 12.6 \, \text{m/s} )
- Force applied in the opposite direction, ( F = -200 \, \text{N} ) (negative because it opposes the motion)
Steps to Calculate Time
- Calculate the Acceleration:
Using Newton’s second law, we can find the acceleration caused by the applied force:
[
F = m \cdot a \quad \Rightarrow \quad a = \frac{F}{m}
]
Substituting the values:
[
a = \frac{-200 \, \text{N}}{50.0 \, \text{kg}} = -4.0 \, \text{m/s}^2
]
The negative sign indicates that the acceleration is in the opposite direction to the motion (deceleration). - Use Kinematic Equation to Find Time:
We can use the following kinematic equation that relates initial velocity, final velocity, acceleration, and time:
[
v_f = v_i + a \cdot t
]
Rearranging for time ( t ):
[
t = \frac{v_f – v_i}{a}
]
Substituting the known values:
[
t = \frac{12.6 \, \text{m/s} – 18.2 \, \text{m/s}}{-4.0 \, \text{m/s}^2} = \frac{-5.6 \, \text{m/s}}{-4.0 \, \text{m/s}^2} = 1.4 \, \text{s}
]
Conclusion
The force was applied for a duration of 1.4 seconds. This process illustrates the interaction between force, mass, and acceleration in a straightforward manner. Understanding these relationships is crucial in fields such as physics and engineering, where forces act on moving objects, affecting their velocities and motions. The application of Newton’s laws and kinematic equations helps analyze real-world scenarios, such as vehicles braking or objects decelerating due to friction or air resistance.