How to Calculate Force, Distance Traveled, and Work Done on a Moving Cart

Problem: A cart and its rider, with a total mass of 85 kg, emerge from a downhill track onto a horizontal straight track with an initial speed of 37 m/s. If a force slows them to a stop at a constant rate of 2.0 m/s², find: (a) the magnitude of force F required, (b) the distance d they travel while slowing down, and (c) the work W done on them by the force.

Solution:

Part (a): Finding the Force F Required

1. Apply Newton's Second Law: Force (F) = Mass (m) × Acceleration (a)
2. Identify the Values:

• • Mass (m) = 85 kg
  • Acceleration (a) = -2.0 m/s² (negative because it is slowing down)

1. Calculate the Force: F = 85 kg × (-2.0 m/s²) = -170 N

• • The negative sign indicates that the force is acting in the opposite direction to the motion.

Part (b): Finding the Distance d Traveled While Slowing Down

1. Use the Kinematic Equation: v² = v₀² + 2aΔx
2. Identify the Values:

• • Final velocity (v) = 0 m/s (since the cart comes to a stop)
  • Initial velocity (v₀) = 37 m/s
  • Acceleration (a) = -2.0 m/s²

1. Calculate the Distance: 0 = (37 m/s)² + 2(-2.0 m/s²)Δx Solving for Δx, we get Δx = 342.25 m

• • The distance traveled while slowing is 342.25 meters.

Part (c): Finding the Work W Done on Them by the Force

1. Use the Work Formula: Work (W) = Force (F) × Distance (Δx)
2. Identify the Values:

• • Force (F) = -170 N
  • Distance (Δx) = 342.25 m

1. Calculate the Work: W = -170 N × 342.25 m = -58182.5 J

• • The negative sign indicates that the force and displacement are in opposite directions.

Key Takeaways:

1. Newton's Second Law: Force causes acceleration, and they always have the same direction.
2. Kinematic Equations: Useful for finding distances when dealing with constant acceleration.
3. Work Done: Work is the dot product of force and displacement, indicating the relationship between force, distance, and the angle between them. Negative work means the force is opposite to the displacement direction.

Remember to always pay attention to the direction of forces and motions to avoid sign errors in your calculations.