(a) If a flea can jump straight up to a height of 0.440 m, what is its initial speed as it leaves the ground?

Verified step by step guidance

1. Identify the relevant kinematic equation for uniformly accelerated motion that relates initial velocity, acceleration, and displacement. The appropriate equation here is: \( v_f^2 = v_i^2 + 2a\Delta x \), where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, \( a \) is the acceleration, and \( \Delta x \) is the displacement.
2. Set the final velocity \( v_f \) to zero, since the flea comes to a temporary stop at the peak of its jump.
3. Recognize that the acceleration \( a \) is due to gravity, which should be taken as \( -9.8 \, \text{m/s}^2 \) (negative because it acts downward while the flea jumps upward).
4. Substitute the known values into the equation: \( 0 = v_i^2 - 2(9.8)(0.440) \).
5. Solve the equation for \( v_i \) (initial velocity) by isolating \( v_i \) on one side of the equation.

Key Concepts

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this case, we can use the equation that relates initial velocity, final velocity, acceleration, and displacement to find the initial speed of the flea. The relevant equation is v^2 = u^2 + 2as, where v is the final velocity (0 m/s at the peak), u is the initial velocity, a is the acceleration (due to gravity), and s is the height (0.440 m).

Acceleration due to Gravity

Acceleration due to gravity is the rate at which an object accelerates towards the Earth when in free fall, approximately 9.81 m/s². This value is crucial for calculating the initial speed of the flea as it jumps, as it will decelerate at this rate until it reaches its maximum height. Understanding this concept allows us to apply the correct value in our kinematic equations.

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to the force of gravity. In this scenario, the flea's jump can be analyzed as a vertical projectile motion, where the only force acting on it after takeoff is gravity. This concept helps in understanding how the initial speed affects the maximum height achieved during the jump.