A constant force is exerted for a short time

Question

interval on a cart that is initially at rest on
an air track. This force gives the cart a
certain final speed. Suppose we repeat the
experiment but, instead of starting from rest,
the cart is already moving with constant speed
in the direction of the force at the moment we
begin to apply the force.
After we exert the same constant force for
the same short time interval, the increase in
the cart’s speed

1. is equal to two times its initial speed.
2. is equal to four times its initial speed.
3. cannot be determined from the information provided.
4. is equal to the square of its initial speed.
5. is the same as when it started from rest.

Answer 1

To analyze the situation, let's apply the basic principles of physics, specifically concepts from kinematics and dynamics.

When a constant force is exerted on an object, the acceleration of that object can be determined using Newton's second law, \( F = ma \). The acceleration \( a \) will depend on the force applied \( F \) and the mass \( m \) of the cart.

Experiment 1: Cart at Rest

1. Initially, the cart is at rest, so its initial speed \( u = 0 \).
2. After a short time \( t \), the force \( F \) results in an acceleration \( a \), which increases the velocity of the cart. Given constant acceleration, the final speed \( v \) can be expressed as: \[ v = u + at = 0 + at = at \] This increase in speed is directly dependent on the force applied and the mass of the cart.

Experiment 2: Cart Already in Motion

1. Now the cart is already moving with some initial speed \( u \) in the direction of the force.
2. When we apply the same force \( F \) for the same time interval \( t \), we still get the same acceleration \( a \): \[ a = \frac{F}{m} \]
3. The final speed in this scenario, considering the initial speed, is given by: \[ v = u + at \] Here, the increase in speed \( \Delta v \) is: \[ \Delta v = at \]

In both scenarios, the increase in speed \( \Delta v \) under the influence of the same constant force and the same time interval is the same, which corresponds to \( at \).

Conclusion:

Thus, the increase in the cart's speed when starting from rest is the same as the increase in speed when starting from an initial velocity.

The answer is:

5. is the same as when it started from rest.