During a crash test, automakers determine the impulse to be 150 N·s on a 10.0 kg test dummy. What is the crash dummy’s speed just before the collision occurred?
Understand the Problem
The question is asking for the calculation of the crash dummy's speed just before a collision, given the impulse and the mass of the dummy. The impulse-momentum theorem relates impulse to the change in momentum, allowing us to solve for speed.
Answer
The crash dummy's speed just before the collision occurred is $15 \, \text{m/s}$.
Answer for screen readers
The crash dummy's speed just before the collision occurred is $15 , \text{m/s}$.
Steps to Solve
- Understand the Impulse-Momentum Theorem
The impulse-momentum theorem states that impulse equals the change in momentum. This can be expressed mathematically as: $$ J = \Delta p $$ where ( J ) is the impulse and ( \Delta p ) is the change in momentum.
- Relate Momentum to Velocity
Momentum (( p )) is defined as the product of mass (( m )) and velocity (( v )): $$ p = mv $$ Thus, the change in momentum can also be expressed in terms of initial and final velocities. If we assume that the dummy starts from rest before the collision, then: $$ \Delta p = mv - 0 = mv $$
- Set Up the Equation
From the impulse-momentum theorem: $$ J = mv $$
- Substitute Known Values
Plug in the values for the impulse and the mass of the dummy: $$ 150 , \text{N} \cdot \text{s} = 10.0 , \text{kg} \cdot v $$
- Solve for Velocity
To find ( v ), rearrange the equation: $$ v = \frac{J}{m} = \frac{150 , \text{N} \cdot \text{s}}{10.0 , \text{kg}} $$
- Calculate the Speed
Now calculate ( v ): $$ v = 15 , \text{m/s} $$
The crash dummy's speed just before the collision occurred is $15 , \text{m/s}$.
More Information
Impulse is a vector quantity that represents the change in momentum of an object. The relationship between impulse and momentum is crucial in understanding collisions in physics.
Tips
- Neglecting Initial Velocity: Forgetting to account for the initial velocity can lead to errors, but in this case, it’s stated that the dummy starts from rest.
- Incorrect Units: Ensure consistency in units. Always check that mass is in kilograms and impulse in Newton-seconds.
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