Question

A \(3.00-\mathrm{kg}\) ball of clay with a speed of \(21.0 \mathrm{~m} / \mathrm{s}\) is thrown against a wall and sticks to the wall. What is the magnitude of the impulse exerted on the ball?

Short answer

Answer: The magnitude of the impulse exerted on the ball is 63.0 kg m/s.

Step-by-step solution

Identify the given information and write down the formula for impulse

The given information is: - Mass of ball, m = 3.00 kg - Initial velocity, v_initial = 21.0 m/s - Final velocity, v_final = 0 m/s (sticks to the wall) The impulse is calculated using the following formula: Impulse = m * (v_final - v_initial)

Calculate the change in velocity

In this step, we'll find the change in velocity as follows: Change in velocity = v_final - v_initial Change in velocity = 0 m/s - 21.0 m/s Change in velocity = -21.0 m/s (since it's just the magnitude, we'll consider the absolute value)

Calculate the impulse

Now, using the impulse formula, we'll find the magnitude of the impulse exerted on the ball: Impulse = m * (v_final - v_initial) Impulse = 3.00 kg * |-21.0 m/s| Impulse = 3.00 kg * 21.0 m/s Impulse = 63.0 kg m/s So, the magnitude of the impulse exerted on the ball is 63.0 kg m/s.

Key concepts

Conservation of Momentum

Understanding the conservation of momentum is fundamental when dealing with collisions and impulses in physics. The principle articulates that in a closed system with no external forces, the total momentum before an event is equal to the total momentum after the event.

When our aforementioned clay ball strikes the wall, the system consists of the ball and the wall. Since the ball sticks to the wall, we can imagine that, momentarily, they become a single system. If the wall were free to move, the combined system of the wall and clay would move with a momentum equal to that of the clay ball before the collision because momentum is conserved. However, since walls are typically anchored firmly, they don’t move, and thus absorb all the momentum delivered by the ball.

In a theoretical context where the wall could move, if we knew its mass and could measure its post-collision velocity, we could use conservation of momentum to calculate those values, because the momentum before the collision (entirely from the ball) would be equal to the momentum after (shared between the ball and the wall).

Collision

Collisions involve objects coming into contact with each other, which provides a practical depiction of momentum and impulse. In physics, there are two main types of collisions: elastic and inelastic. Elastic collisions are those in which kinetic energy is conserved—objects bounce off each other with no loss of total kinetic energy. Inelastic collisions, in contrast, result in some loss of kinetic energy, and sometimes objects stick together, like the clay ball sticking to the wall.

In our exercise, we are examining a perfectly inelastic collision, where the clay ball doesn't bounce off but rather adheres to the wall. Here, the ball’s final velocity becomes zero—not because it doesn’t move, but because it can't move independently of the wall. The impulse exerted on the clay ball can indicate the magnitude of force during the impact and is a direct result of the change in the ball’s momentum.

Change in Velocity

Velocity can change in two ways: a change in speed, or a change in direction, or both. In the context of our exercise involving the clay ball, the velocity change is exclusively concerning the ball’s speed, which reduces from the initial 21.0 m/s to 0 m/s as it sticks to the wall.

The magnitude of this velocity change is substantial because it signifies a total stop; the ball’s motion ceases in the direction it was initially moving. This abrupt halt is also reflected in the calculation of impulse, which equals the product of the mass and the velocity change. Notice that only the speed's magnitude is considered in the calculation because impulse is a vector quantity and has both magnitude and direction; the negative sign in the change of velocity represents a direction opposite to the initial motion but does not affect the magnitude of the impulse.