A 45 g projectile explodes into three pieces: a 20 g piece with velocity 25 î m/s, a 15 g piece with velocity −10 î + 10ĵ m/s, and a 10 g piece with velocity −15 î − 20ĵ m/s. What was the projectile's velocity just before the explosion?

Verified step by step guidance

1. Step 1: Understand the principle of conservation of momentum. In the absence of external forces, the total momentum of a system before and after an event (like an explosion) remains constant. Here, the total momentum of the projectile before the explosion equals the total momentum of its fragments after the explosion.
2. Step 2: Write the expression for the total momentum of the system before the explosion. Let the velocity of the projectile before the explosion be \( \vec{v}_i \). The mass of the projectile is \( m = 45 \ \text{g} = 0.045 \ \text{kg} \). The total momentum before the explosion is \( \vec{p}_i = m \vec{v}_i \).
3. Step 3: Write the expression for the total momentum of the system after the explosion. The momentum of each fragment is the product of its mass and velocity. For the three fragments, the total momentum after the explosion is: \( \vec{p}_f = m_1 \vec{v}_1 + m_2 \vec{v}_2 + m_3 \vec{v}_3 \), where \( m_1 = 0.020 \ \text{kg}, \ m_2 = 0.015 \ \text{kg}, \ m_3 = 0.010 \ \text{kg} \), and their respective velocities are given.
4. Step 4: Substitute the given velocities into the expression for \( \vec{p}_f \). Using the velocities \( \vec{v}_1 = 25 \hat{i} \ \text{m/s}, \ \vec{v}_2 = -10 \hat{i} + 10 \hat{j} \ \text{m/s}, \ \vec{v}_3 = -15 \hat{i} - 20 \hat{j} \ \text{m/s} \), calculate the total momentum after the explosion: \( \vec{p}_f = (0.020)(25 \hat{i}) + (0.015)(-10 \hat{i} + 10 \hat{j}) + (0.010)(-15 \hat{i} - 20 \hat{j}) \).
5. Step 5: Set \( \vec{p}_i = \vec{p}_f \) to solve for \( \vec{v}_i \). Since \( \vec{p}_i = m \vec{v}_i \), rearrange to find \( \vec{v}_i = \frac{\vec{p}_f}{m} \). Substitute the total momentum \( \vec{p}_f \) and the mass \( m = 0.045 \ \text{kg} \) to determine the initial velocity of the projectile.

Key Concepts

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In this case, the momentum of the projectile before it explodes must equal the combined momentum of the three pieces after the explosion, allowing us to calculate the initial velocity of the projectile.

Momentum Calculation

Momentum is calculated as the product of mass and velocity (p = mv). For each piece of the projectile after the explosion, we can calculate its momentum by multiplying its mass by its velocity vector. Summing these momenta will give us the total momentum after the explosion, which we can then set equal to the initial momentum to find the projectile's velocity.

Vector Addition

Since momentum is a vector quantity, we must consider both the magnitude and direction of the velocities involved. Vector addition is used to combine the momentum vectors of the pieces after the explosion. This involves adding the components of the vectors separately in the x and y directions to find the resultant momentum vector, which is essential for determining the initial velocity of the projectile.