In this scenario, let's break down the effects of applying the same constant force on the cart under two different initial conditions: starting from rest versus starting with an initial constant speed.
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Starting from Rest:
- When the cart is initially at rest and we apply a constant force \( F \) for a time interval \( t \), the acceleration \( a \) of the cart can be calculated using Newton's second law: \( F = ma \), where \( m \) is the mass of the cart.
- The acceleration \( a \) is given by \( a = \frac{F}{m} \).
- The increase in speed \( \Delta v \) can be calculated using the formula: \[ \Delta v = a \cdot t = \frac{F}{m} \cdot t. \]
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Starting with Constant Speed:
- In the second scenario, the cart is already moving with a constant speed \( v_0 \) in the direction of the force when the same constant force \( F \) is applied for the same time interval \( t \).
- The same initial acceleration \( a \) applies, so the increase in speed \( \Delta v \) due to the same force applied over the same time is still \[ \Delta v = \frac{F}{m} \cdot t. \]
- As a result, the final speed after the force is applied is: \[ v_f = v_0 + \Delta v. \]
Conclusion:
The increase in the cart's speed, \( \Delta v \), due to the constant force applied over the same time interval will be the same in both cases, since it depends solely on the force and mass of the cart.
However, the total final speed will be higher in the second case (starting with initial speed \( v_0 \)) compared to the first case (starting from rest).
Thus, in both situations, the increase in speed due to the force is the same, but the total speed of the cart after the force is applied will be greater when starting from an initial speed.
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