Question

ben is learning to play tennis. today he is practicing his service motion. bens practice routine is to toss a tennis ball straight up in the air and then catch it when it returns to the point at which it was tossed. on one practice toss, the ball leaves bens hand at a speed of 9.8 meters per second. let v(t) denote the velocity of the ball (in meters per second) t seconds after it is tossed on the occasion mentioned above. the equation below relates v(t) and t. v(t) = 9.8 - 9.8t what is the domain of this relationship? a. 0, 2 b. 0, ∞) c. 0, 9.8

Explanation:

Step1: Understand the context

The problem is about the velocity of a tossed tennis ball over time. Time \( t \) starts at 0 (when tossed) and the ball is caught when it returns, so we need to find when the ball returns (velocity \( v(t) = 0 \) as it comes back to the starting point).

Step2: Solve for \( t \) when \( v(t) = 0 \)

Given \( v(t)=9.8 - 9.8t \), set \( v(t) = 0 \):

$$ 0 = 9.8 - 9.8t $$

Add \( 9.8t \) to both sides:

$$ 9.8t = 9.8 $$

Divide both sides by \( 9.8 \):

$$ t = 1 $$

Wait, but actually, the ball is tossed up and comes back. The total time in the air: when it's tossed, \( t = 0 \), and when it returns, \( v(t)=0 \) (at the top, but actually when it comes back, the displacement is zero. Wait, maybe I made a mistake. Wait, the velocity function: initial velocity \( 9.8 \) m/s, acceleration \( - 9.8 \) m/s² (gravity). The ball is in the air until it comes back to the starting point. The time to go up and come back: the time to reach the top is when \( v(t)=0 \), which is \( t = 1 \) second, then it comes back down, so total time is \( 2 \) seconds? Wait, no, if you toss a ball up with initial velocity \( v_0 = 9.8 \) m/s, the time to reach the top is \( t = v_0/g = 9.8/9.8 = 1 \) second, then it takes another 1 second to come back down, so total time in the air is \( 2 \) seconds. So \( t \) ranges from \( 0 \) to \( 2 \) seconds (since at \( t = 2 \), it's back to the starting point). So the domain of \( t \) is \( [0, 2] \).

Answer:

A. [0, 2]