Question

The position of a baseball (in ft ) is represented by r(t)=40√2ti+(3+40√2t−16t²)j. Find the arc length of the trajectory (flight) of the baseball. (Let t=0 represent when the ball is hit. Let the j component equal 0 at ground level. Round your answer to one decimal place.) ₹ ft

Answer 1

The total arc length of the baseball's trajectory is calculated by integrating the magnitude of the derivative of the position vector over the time from when the ball is hit until it lands.

Step-by-step explanation:

The question involves calculating the arc length of the path of a baseball using a position vector function r(t).

The arc length from a vector function can be computed by the integral of the magnitude of the derivative of the position vector with respect to time, t. Applying this for the given function:

r(t) = 40√2ti + (3 + 40√2t - 16t²)j,

we find the derivatives of the i and j components respectively:

dx/dt = d(40√2t)/dt = 40√2,

dy/dt = d(3 + 40√2t - 16t²)/dt = 40√2 - 32t.

Then we calculate the magnitude of the derivative (√((dx/dt)² + (dy/dt)²)) and integrate over the appropriate interval of t to find the total arc length.

Since the position vector's j component represents height and is zero at ground level, we need to identify the interval from the time the ball is hit to when it lands again (when the j component is zero).