A car (mass 880 kg) is traveling with a speed of 44 mi/h. A bug (mass 7.9 mg) is traveling in the opposite direction. What speed would the bug need in order to slow down the truck by one mile per hour in a big bug splash (i.e., final speed 43 mi/h)?
A car (mass 880 kg) is traveling with a speed of 44 mi/h. A bug (mass 7.9 mg) is traveling in the opposite direction. What speed would the bug need in order to slow down the truck by one mile per hour in a big bug splash (i.e., final speed 43 mi/h)?
To find the speed at which the bug needs to travel in order to slow down the car by one mile per hour, we can use the law of conservation of linear momentum. According to this law, the total linear momentum before and after a collision remains the same.
The linear momentum of an object is given by the product of its mass and velocity. Therefore, initially, the total linear momentum of the system (car + bug) is the sum of their individual momenta.
Let's denote:
- m_car as the mass of the car (880 kg)
- v_car as the initial velocity of the car (44 mi/h)
- m_bug as the mass of the bug (7.9 mg = 7.9 × 10^(-6) kg)
- v_bug as the required velocity of the bug
The initial linear momentum of the car is given by m_car * v_car, and the initial linear momentum of the bug is given by m_bug * (-v_bug) since the bug is traveling in the opposite direction.
The final linear momentum of the system, after the collision, will be dictated by the new velocity of the car. We know that the final velocity of the car is 43 mi/h, so the final linear momentum of the car is m_car * 43.
According to the law of conservation of linear momentum, we can set up the equation:
Initial linear momentum of the system = Final linear momentum of the system
(m_car * v_car) + (m_bug * (-v_bug)) = m_car * 43
Plugging in the values, we get:
(880 kg * (44 mi/h)) + ((7.9 × 10^(-6) kg) * (-v_bug)) = 880 kg * 43
Simplifying further:
38,720 kg⋅mi/h - (7.9 × 10^(-6) kg * v_bug) = 37,840 kg⋅mi/h
Now, let's isolate v_bug by rearranging the equation:
(7.9 × 10^(-6) kg * v_bug) = 38,720 kg⋅mi/h - 37,840 kg⋅mi/h
(7.9 × 10^(-6) kg * v_bug) = 880 kg⋅mi/h
v_bug = (880 kg⋅mi/h) / (7.9 × 10^(-6) kg)
Calculating this value, we find:
v_bug ≈ 1.11 × 10^8 mi/h
Therefore, the bug would need to travel at approximately 1.11 × 10^8 miles per hour in the opposite direction to slow down the car by one mile per hour.