A 2.0 kg projectile with initial velocity v = 8.0 î m/s experiences the variable force F = -2.0t î + 4.0t² ĵ N, where t is in s. What is the projectile's speed at t = 2.0 s?

Verified step by step guidance

1. Step 1: Begin by identifying the relationship between force and acceleration using Newton's second law, \( F = ma \). Since the mass \( m \) of the projectile is given as 2.0 kg, the acceleration \( a \) can be calculated as \( a = \frac{F}{m} \). The force \( F \) is given as \( F = -2.0t \hat{i} + 4.0t^2 \hat{j} \). Divide each component of \( F \) by the mass to find the acceleration vector.
2. Step 2: Integrate the acceleration vector with respect to time to find the velocity vector. The acceleration components are \( a_x = \frac{-2.0t}{2.0} \) and \( a_y = \frac{4.0t^2}{2.0} \). Integrating \( a_x \) and \( a_y \) with respect to \( t \) gives the velocity components \( v_x \) and \( v_y \). Remember to include the initial velocity \( v_x = 8.0 \) m/s in the \( \hat{i} \) direction.
3. Step 3: Evaluate the velocity components \( v_x \) and \( v_y \) at \( t = 2.0 \) s. Substitute \( t = 2.0 \) into the expressions for \( v_x \) and \( v_y \) obtained from the integration step.
4. Step 4: Calculate the magnitude of the velocity vector (speed) using the formula \( v = \sqrt{v_x^2 + v_y^2} \). Substitute the values of \( v_x \) and \( v_y \) at \( t = 2.0 \) s into this formula.
5. Step 5: The result from the previous step gives the projectile's speed at \( t = 2.0 \) s. Ensure all units are consistent throughout the calculation (e.g., meters per second for velocity).

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed mathematically as F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing how forces affect the motion of the projectile.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as velocity, acceleration, and displacement. In this problem, kinematic equations can be used to determine the projectile's speed at a specific time by integrating the acceleration derived from the force.

Integration of Force to Find Velocity

To find the velocity of an object when a variable force is acting on it, one must integrate the force over time to obtain the change in momentum, which relates to velocity. The force given in the problem is a function of time, so integrating this force will yield the impulse, which can then be used to find the final velocity of the projectile at t = 2.0 s.