A 1420-kg car moving at 5.28 m/s is initially traveling north. After completing a 90° right-hand turn in 4.60 s, the inattentive operator drives into a tree, which stops the car in 350 ms. What is the magnitude of the impulse delivered to the car (a) during the turn and (b) during the collision? What average force acts on the car (c) during the tum and (d) during the collision?

To tackle this problem, we need to break it down into two parts: the impulse during the turn and the impulse during the collision. Impulse is defined as the change in momentum of an object, which can be calculated using the formula: Impulse = Change in Momentum = Final Momentum - Initial Momentum. We will also use the relationship between impulse and force, which states that impulse is equal to the average force multiplied by the time over which the force acts: Impulse = Average Force × Time.

Impulse During the Turn

First, let's find the impulse delivered to the car during the turn. The car is initially moving north at a speed of 5.28 m/s. After making a 90° right-hand turn, it will be moving east. We need to determine the change in momentum.

Calculating Initial and Final Momentum

The initial momentum of the car can be calculated as:

• Initial Momentum (p_initial) = mass × velocity = 1420 kg × 5.28 m/s = 7497.6 kg·m/s (north)

After the turn, the car's velocity is now directed east at the same speed:

• Final Momentum (p_final) = mass × velocity = 1420 kg × 5.28 m/s = 7497.6 kg·m/s (east)

Finding the Change in Momentum

Since the momentum is a vector quantity, we need to consider the direction. The change in momentum can be calculated using vector subtraction:

• Change in Momentum = Final Momentum - Initial Momentum
• Change in Momentum = (0, 7497.6) - (0, -7497.6) = (7497.6, 7497.6) kg·m/s

To find the magnitude of the change in momentum:

• Magnitude = √((7497.6)^2 + (7497.6)^2) = 7497.6√2 ≈ 10610.4 kg·m/s

Calculating Impulse

The impulse delivered to the car during the turn is approximately 10610.4 kg·m/s.

Average Force During the Turn

Now, we can find the average force acting on the car during the turn. We know the impulse and the time duration of the turn:

• Time = 4.60 s

Using the impulse formula:

• Impulse = Average Force × Time
• Average Force = Impulse / Time = 10610.4 kg·m/s / 4.60 s ≈ 2302.3 N

Impulse During the Collision

Next, we need to calculate the impulse delivered to the car during the collision with the tree. The car comes to a complete stop, so its final velocity is 0 m/s.

Calculating Change in Momentum for the Collision

The initial momentum before the collision is:

• Initial Momentum = 7497.6 kg·m/s (moving east)

The final momentum after the collision is:

• Final Momentum = 0 kg·m/s

The change in momentum during the collision is:

• Change in Momentum = Final Momentum - Initial Momentum = 0 - 7497.6 = -7497.6 kg·m/s

The magnitude of the impulse during the collision is 7497.6 kg·m/s.

Average Force During the Collision

To find the average force acting on the car during the collision, we need the time duration of the collision:

• Time = 350 ms = 0.350 s

Using the impulse formula again:

• Average Force = Impulse / Time = 7497.6 kg·m/s / 0.350 s ≈ 21421.7 N

Summary of Results

To summarize:

• Impulse during the turn: 10610.4 kg·m/s
• Average force during the turn: 2302.3 N
• Impulse during the collision: 7497.6 kg·m/s
• Average force during the collision: 21421.7 N

This analysis shows how the concepts of momentum and impulse are applied in real-world scenarios, such as vehicle dynamics during turns and collisions. Understanding these principles is crucial for fields like physics, engineering, and safety design.