A hockey player is standing on his skates on a frozen pond when an opposing player, moving with a uniform speed of 14 m/s, skates by with the puck. After 3.5 s, the first player makes up his mind to chase his opponent.
(a) If he accelerates uniformly at 4.0 m/s2, how long does it take him to catch his opponent? (Assume the player with the puck remains in motion at constant speed.)
s
(b) How far has he traveled in this time?
m
For this question t^2+ (what s)t+ what s^2=0
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(a) To calculate the time it takes for the first player to catch his opponent, we can use the formula:
\(d = v_0t + \frac{1}{2}at^2\)
where \(d\) is the distance traveled, \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time.
In this case, the initial velocity of the first player is 0 m/s since he is initially standing still. The acceleration is 4.0 m/s², and the time we are trying to find is \(t\).
The distance the first player needs to catch up is the same distance covered by the second player who is moving with a speed of 14 m/s for 3.5 s, given by:
\(d = v \cdot t\)
Substituting the known values:
\(d = 14 \cdot 3.5\)
\(d = 49\) m
Now we can use the distance and acceleration to find the time it takes for the first player to catch up:
\(49 = 0 \cdot t + \frac{1}{2} \cdot 4.0 \cdot t^2\)
\(\frac{1}{2} \cdot 4.0 \cdot t^2 = 49\)
\(2.0 \cdot t^2 = 49\)
\(t^2 = 24.5\)
\(t = \sqrt{24.5}\)
\(t \approx 4.95\) s
Therefore, it takes the first player approximately 4.95 seconds to catch his opponent.
(b) To find out how far the first player has traveled in this time, we can use the formula:
\(d = v_0t + \frac{1}{2}at^2\)
where \(d\) is the distance traveled, \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time.
Substituting the known values:
\(d = 0 \cdot 4.95 + \frac{1}{2} \cdot 4.0 \cdot (4.95)^2\)
\(d = 0 + \frac{1}{2} \cdot 4.0 \cdot (4.95)^2\)
\(d \approx 48.62\) m
Therefore, the first player has traveled approximately 48.62 meters in this time.
To solve this problem, we can break it down into two parts: finding the time it takes for the first player to catch his opponent and calculating the distance traveled during that time.
(a) Finding the time it takes to catch the opponent:
The opposing player is moving at a uniform speed of 14 m/s. The first player, who initially stands still, needs to accelerate uniformly at 4.0 m/s^2 to catch his opponent. We can use the equation of motion to find the time it takes for the first player to catch the opponent. The equation is:
distance = initial velocity * time + (1/2) * acceleration * time^2
Since the initial velocity of the first player is zero, the equation simplifies to:
distance = (1/2) * acceleration * time^2
Rearranging the equation to solve for time:
time = √((2 * distance) / acceleration)
The distance is the distance between the first player and the opposing player at the moment the first player starts chasing. In this case, the distance is not given; it needs to be calculated using the constant speed and time.
The opposing player moves for 3.5 seconds before the first player starts chasing. So, the distance traveled by the opposing player in 3.5 seconds is:
distance = speed * time = 14 m/s * 3.5 s
Now we can substitute the values into our equation to calculate the time:
time = √((2 * distance) / acceleration) = √((2 * (14 m/s * 3.5 s)) / 4.0 m/s^2)
Calculating this expression will give us the time it takes for the first player to catch the opponent.
(b) Finding the distance traveled by the first player:
Once we have the time it takes for the first player to catch the opponent, we can calculate the distance traveled by the first player during that time.
The equation of motion for distance traveled is:
distance = initial velocity * time + (1/2) * acceleration * time^2
In this case, the initial velocity is zero because the first player starts from rest. The time is the one we calculated in part (a), and the acceleration is given as 4.0 m/s^2.
Substituting the values into the equation, we can calculate the distance traveled by the first player:
distance = (1/2) * acceleration * time^2
Calculating this expression will give us the distance traveled by the first player.
To summarize:
(a) Calculate the time it takes for the first player to catch the opponent using the equation time = √((2 * distance) / acceleration).
(b) Calculate the distance traveled by the first player during that time using the equation distance = (1/2) * acceleration * time^2.
d1 = V1*t = 14m/s * 3.5s. = 49 m. Head-start.
a. d2 = 49 + V1*t.
0.5a*t^2 = 49 + V1*t.
0.5*4*t^2 = 49 + 14*t.
2t^2-14t-49 = 0.
Use Quadratic Formula.
t = 9.56 s.
b. d = 0.5a*t^2.