Question

An 83.0 -kg running back leaps straight ahead toward the end zone with a speed of \(6.50 \mathrm{~m} / \mathrm{s}\). A 115 -kg linebacker, keeping his feet on the ground, catches the running back and applies a force of \(900 . \mathrm{N}\) in the opposite direction for 0.750 s before the running back's feet touch the ground. a) What is the impulse that the linebacker imparts to the running back? b) What change in the running back's momentum does the impulse produce? c) What is the running back's momentum when his feet touch the ground? d) If the linebacker keeps applying the same force after the running back's feet have touched the ground, is this still the only force acting to change the running back's momentum?

Short answer

Answer: The final momentum of the running back when his feet touch the ground is -135.5 kg.m/s. When the running back's feet touch the ground, other forces such as gravity (weight) act on him in addition to the linebacker's force.

Step-by-step solution

(Step 1: Calculate the initial momentum of the running back)

First, we can find the initial momentum of the running back using the formula: \(momentum = mass \times velocity\). With \(mass=83.0\,\text{kg}\) and \(velocity=6.50\,\text{m/s}\): \(initial\_momentum = 83.0 \,\text{kg} \times 6.50\,\text{m/s} = 539.5\,\text{kg}\cdot\text{m/s}\)

(Step 2: Calculate the impulse delivered by the linebacker)

Impulse is defined as the product of force and time: \(Impulse = Force \times Time\). In this case, we have \(force = -900\,\text{N}\) (negative sign as it acts in the opposite direction of the running back's motion), and \(time = 0.750\,\text{s}\). Calculate the impulse: \(Impulse = -900\,\text{N} \times 0.750\,\text{s} = -675\,\text{kg}\cdot\text{m/s}\) The impulse that the linebacker imparts to the running back is -675 kg.m/s.

(Step 3: Determine the change in momentum of the running back)

The change in momentum is equal to the impulse delivered by the linebacker: \(\Delta momentum = Impulse\) Thus, the change in the running back's momentum is -675 kg.m/s.

(Step 4: Calculate the final momentum of the running back)

We know that the final momentum of the running back is the sum of his initial momentum and the change in momentum (impulse provided by the linebacker): \(final\_momentum = initial\_momentum + \Delta momentum\) \(final\_momentum = 539.5\,\text{kg}\cdot\text{m/s} - 675\,\text{kg}\cdot\text{m/s} = -135.5\,\text{kg}\cdot\text{m/s}\) The running back's momentum when his feet touch the ground is -135.5 kg.m/s.

(Step 5: Determine the forces acting on the running back when his feet touch the ground)

When the running back's feet touch the ground, the force of gravity (weight) also acts on him, along with the opposing force applied by the linebacker. The force of gravity acts vertically downward while the linebacker's force still acts horizontally. Therefore, the linebacker's force is not the only force acting on the running back when his feet have touched the ground.

Key concepts

Force

When discussing impulse and momentum, one of the key players is force. Force is the interaction that causes an object to change its motion. It's what acts to alter the velocity, hence affecting an object's momentum. In our given exercise, the linebacker applies a force of 900 N against the running back. This opposing force acts to decelerate the running back's speed.
Force is measured in newtons (N) in the International System of Units (SI), and it can be calculated using Newton's second law, which states that force equals mass times acceleration: \[ F = m imes a \] Here, force acts in the opposite direction of the running back’s initial motion, making it negative in our calculations. Knowing the time the force is applied is crucial, as this will help determine the impulse - which is the product of force and the duration it was in effect.

Change in momentum

Momentum is the quantity of motion an object has, and change in momentum is a fundamental concept when studying motion dynamics. This change, often referred to as "impulse" in physics, occurs when a force is applied over a period of time.
According to the impulse-momentum theorem, the change in momentum of an object is equal to the impulse applied to it. In mathematical terms, it's given by:\[ \Delta p = F \times t \] Where \( \Delta p \) is the change in momentum, \( F \) is the applied force, and \( t \) is the time duration during which the force acts. In our example, the linebacker's force results in a momentum change of \(-675 \, \text{kg} \cdot \text{m/s}\), which denotes a decrease since the force is opposite to the running back's motion. Understanding this relationship helps in solving problems where you need to find the final state of an object's motion after a force has been applied.

Linear momentum

Linear momentum is a vector quantity that depends on both the mass and velocity of an object. It plays a crucial role in understanding motion in a straight line. The formula for linear momentum \( p \) is straightforward: \[ p = m \times v \] Where \( m \) is the mass in kilograms (kg), and \( v \) is the velocity in meters per second (m/s). In our scenario, the running back initially has a linear momentum of \( 539.5 \, \text{kg} \cdot \text{m/s} \) before the linebacker applies the force.
Linear momentum provides insight into how the moving running back can continue in motion until acted upon by another force, reflecting Newton's first law of motion. When the linebacker intercepts, the momentum changes. The initial and changed momenta help predict how much motion remains or is altered following the external intervention.