An 18-kg child is riding in a child-restraint chair, securely fastened to the seat of a car (Fig. 4–69). Assume the car has speed 45 km/h when it hits a tree and is brought to rest in 0.20 s. Assuming constant deceleration during the collision, estimate the net horizontal force F that the straps of the restraint chair exert on the child to hold her in the chair. <IMAGE>

Verified step by step guidance

1. Convert the initial speed of the car from km/h to m/s. Use the conversion factor: 1 km/h = 1000 m / 3600 s. Thus, \( v_0 = 45 \; \text{km/h} \) can be converted to \( v_0 \; \text{(m/s)} \).
2. Identify the final velocity \( v_f \) of the car and the child after the collision. Since the car comes to rest, \( v_f = 0 \; \text{m/s} \).
3. Use the kinematic equation to calculate the acceleration \( a \): \( v_f = v_0 + a t \). Rearrange to solve for \( a \): \( a = \frac{v_f - v_0}{t} \), where \( t = 0.20 \; \text{s} \) is the time taken to stop.
4. Apply Newton's second law of motion, \( F = m a \), to calculate the net horizontal force \( F \) exerted on the child. Here, \( m = 18 \; \text{kg} \) is the mass of the child, and \( a \) is the acceleration calculated in the previous step.
5. Substitute the values of \( m \) and \( a \) into the formula \( F = m a \) to find the magnitude of the net horizontal force exerted by the straps of the restraint chair on the child.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). In this scenario, the child experiences a deceleration when the car collides with the tree, and the net force exerted by the restraint system must counteract this deceleration to keep the child safely in place.

Deceleration

Deceleration is the rate at which an object slows down, which is a negative acceleration. In this case, the car comes to a stop from a speed of 45 km/h in 0.20 seconds, resulting in a specific deceleration that can be calculated. This deceleration is crucial for determining the net force acting on the child during the collision.

Impulse and Momentum

Impulse is the change in momentum of an object when a force is applied over a period of time. The momentum of the child must be considered as the car stops suddenly, and the impulse experienced by the child due to the net force from the restraint system can be calculated. This relationship helps in understanding how the force exerted by the straps affects the child's motion during the collision.