During a football game, Igor is 8.0 m behind Brian and is running at 7.0 m/ s when Brian catches the ball and starts to accelerate away at 2.8 m/ s 2 from rest. a) Will Igor catch Brian? If so, after how long?
To determine whether Igor will catch Brian, we need to compare their distances from the starting point as a function of time.
Let's assume the starting point is the origin (0 m).
For Igor:
Distance travelled by Igor (dI) = 8.0 m (since he starts 8.0 m behind Brian)
Speed of Igor (vI) = 7.0 m/s
For Brian:
Acceleration of Brian (aB) = 2.8 m/s^2
Initial velocity of Brian (uB) = 0 m/s (he starts from rest)
Distance travelled by Brian (dB) = distance travelled by Igor (dI) + distance ahead of Igor
Using the equations of motion:
For Igor:
dI = vI * tI
For Brian:
dB = uB * tB + (1/2) * aB * tB^2
Since we are comparing their distances at the same time, we can equate dI and dB:
vI * tI = uB * tB + (1/2) * aB * tB^2
Substituting the given values:
7.0 m/s * tI = 0 m/s * tB + (1/2) * 2.8 m/s^2 * tB^2
Simplifying the equation:
7.0 tI = 1.4 tB^2
We can rearrange the equation to compare the times:
tB^2 = (7.0 tI) / 1.4
tB^2 = 5.0 tI
Taking the square root of both sides:
tB = sqrt(5.0) * tI
Since tB > tI in this case (Brian accelerates away), Igor will not catch Brian.
Therefore, Igor will not catch Brian during the football game.
To determine if Igor will catch Brian, we need to calculate the time it takes for Igor to catch up with Brian.
Let's start by calculating the initial velocity of Brian. Since Brian is at rest initially and accelerates at a rate of 2.8 m/s^2, we can use the equation:
v = u + at
Where:
- v is the final velocity (which we want to find)
- u is the initial velocity (which is 0 m/s)
- a is the acceleration (2.8 m/s^2)
- t is the time
Substituting the known values into the equation:
v = 0 + (2.8 m/s^2) * t
v = 2.8t m/s
Next, we'll calculate the distance covered by Brian (assuming he starts at the same time Igor starts running). Since Igor is initially 8.0 m behind Brian, we can use the equation:
distance = speed * time
Substituting the known values into the equation:
distance = 7.0 m/s * t + 8.0 m
Now, we'll set up an equation to find the time it takes for Igor to catch Brian.
7.0t + 8.0 = 2.8t
Simplifying the equation:
7.0t - 2.8t = -8.0
4.2t = -8.0
Dividing both sides of the equation by 4.2:
t = -8.0 / 4.2
t ≈ -1.905
The negative value of t does not make sense in this context, so we'll discard it.
Therefore, Igor will not catch Brian during the football game.