Question

Two objects having different masses fall freely under the influence of gravity from rest and the same initial elevation. Ignoring the effect of air resistance, show that the magnitudes of the velocities of the objects are equal at the moment just before they strike the earth.

Short answer

Both objects have the same velocity \( v = gt \) just before they strike the earth, independent of their masses.

Step-by-step solution

Identify the Key Information

The two objects have different masses but fall from the same height. The falling is influenced only by gravity, and air resistance is ignored.

Apply the Equation for Free Fall

Use the kinematic equation for velocity under constant acceleration: \[ v = u + gt \] where \( u \) is the initial velocity, \( g \) is the acceleration due to gravity, and \( t \) is the time.

Simplify the Equation

Since the initial velocity \( u \) is 0 (because the objects are falling from rest), the equation simplifies to: \[ v = gt \]

Recognize Mass Independence

The equation \( v = gt \) shows that the velocity just before striking the ground depends only on the gravitational acceleration \( g \) and the time \( t \), neither of which depend on the mass of the objects.

Conclude the Result

Since both objects fall from the same height and are under the same gravitational influence for the same time duration, their velocities just before striking the earth must be equal.

Key concepts

Kinematic Equations

To understand how objects move, particularly in free fall, we rely on kinematic equations. These equations describe the motion of objects under uniform acceleration. When an object falls freely under gravity, it experiences constant acceleration. The kinematic equation for velocity is given by: \[ v = u + gt \] Here,

• \( v \) represents the final velocity of the object just before it hits the ground.
• \( u \) is the initial velocity of the object. In free fall from rest, this value is 0.
• \( g \) stands for the acceleration due to gravity, which is approximately \( 9.81 m/s^2 \) on Earth.
• \( t \) is the time the object has been falling.

Since the initial velocity \( u \) is 0, the equation simplifies to: \[ v = gt \]This simple equation helps us determine the velocity of any freely falling object just before it contacts the ground.

Gravity

Gravity is a force that attracts two bodies towards each other. On Earth, gravity gives objects a constant acceleration toward the ground, symbolized by \( g \). The magnitude of this acceleration is roughly \( 9.81 m/s^2 \). This means that if you drop an object, without any other forces like air resistance interfering, its velocity increases by \( 9.81 m/s \) every second.For an object in free fall, gravity is the only force acting upon it. Therefore, any object, regardless of its mass, will experience the same gravitational pull and will accelerate uniformly. This forms the basis of why, in the absence of air resistance, two objects of different masses will fall at the same rate and have the same velocity when they hit the ground.

Mass Independence

One of the fascinating aspects of gravitational acceleration is that it is independent of the object's mass. This means whether an object is heavy or light, like a stone or a feather, as long as they are in a vacuum (where there's no air resistance), they will fall at the same rate.The key kinematic equation, \[ v = gt \], further clarifies this. Both \( g \), the gravitational acceleration, and \( t \), the time of fall, are not influenced by the mass of the object. As a result, the final velocity \( v \) just before the object strikes the ground is also mass-independent.This concept can be quite counterintuitive but is easily observed if you drop different objects in a vacuum chamber. Outside of a vacuum, objects of different masses might fall differently because of air resistance. However, when air resistance is ignored, as in our initial exercise, mass does not play a role in determining the final velocity of freely falling bodies.