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 solve this problem, we need to break it down into two parts: the impulse delivered to the car during the turn and during the collision, as well as the average forces acting on the car during these events. Let's tackle each part step by step.

Impulse During the Turn

Impulse is defined as the change in momentum of an object. The formula for impulse (J) can be expressed as:

J = Δp = m * Δv

where:

• m = mass of the car (1420 kg)
• Δv = change in velocity

Initially, the car is moving north at 5.28 m/s. After completing a 90° right-hand turn, the direction of the car changes, but we need to find the new velocity vector. The car's speed remains the same, but its direction changes to east. Thus, the initial velocity vector can be represented as:

v_initial = (0, 5.28) m/s (north)

v_final = (5.28, 0) m/s (east)

Now, we can calculate the change in velocity:

Δv = v_final - v_initial = (5.28, 0) - (0, 5.28) = (5.28, -5.28) m/s

Next, we find the magnitude of this change in velocity:

|Δv| = √((5.28)^2 + (-5.28)^2) = √(27.8784 + 27.8784) = √(55.7568) ≈ 7.46 m/s

Now we can calculate the impulse:

J = m * |Δv| = 1420 kg * 7.46 m/s ≈ 10563.2 kg·m/s

Impulse During the Collision

When the car collides with the tree, it comes to a stop. The initial velocity before the collision is still 5.28 m/s, and the final velocity is 0 m/s. Thus, the change in momentum during the collision is:

Δv_collision = v_final - v_initial = 0 - 5.28 = -5.28 m/s

Now, we can calculate the impulse during the collision:

J_collision = m * Δv_collision = 1420 kg * (-5.28 m/s) = -7509.6 kg·m/s

Average Force During the Turn

The average force (F_avg) can be calculated using the impulse and the time over which it occurs:

F_avg = J / Δt

For the turn, the time taken is 4.60 seconds:

F_avg_turn = J / Δt_turn = 10563.2 kg·m/s / 4.60 s ≈ 2292.0 N

Average Force During the Collision

For the collision, the time taken is 350 milliseconds, which is 0.350 seconds:

F_avg_collision = J_collision / Δt_collision = -7509.6 kg·m/s / 0.350 s ≈ -21456.0 N

The negative sign indicates that the force is acting in the opposite direction of the car's initial motion.

Summary of Results

To summarize, we have:

• Impulse during the turn: 10563.2 kg·m/s
• Impulse during the collision: -7509.6 kg·m/s
• Average force during the turn: 2292.0 N
• Average force during the collision: -21456.0 N

This analysis shows how the car's momentum changes during both the turn and the collision, along with the forces involved in these processes. Understanding these concepts is crucial in physics, especially in mechanics and dynamics.