A tennis ball is projected vertically upwards from the top of a 55 m cliff by the sea. Its height from the point of projection, s m, ts later is given by
s = 50t - 5t^2
a. when does the tennis ball hit the sea?
b. find the impact velocity of the ball as it enters the sea.
c. find the mean speed of the ball for its entire time of flight.
c. (Vo+V)/2 = (50+60)/2 = 55 m/s.
a. When does the tennis ball hit the sea?
Well, according to the equation, the height (s) is equal to zero when the ball hits the sea. So, let's set the equation equal to zero and solve for t:
0 = 50t - 5t^2
Now, let's rearrange the equation:
5t^2 - 50t = 0
To solve this quadratic equation, we can factor out a common term:
5t(t - 10) = 0
So, either 5t = 0 or t - 10 = 0.
If 5t = 0, then t = 0.
If t - 10 = 0, then t = 10.
Since the time (t) cannot be negative, the tennis ball will hit the sea after 10 seconds.
b. Find the impact velocity of the ball as it enters the sea.
To find the impact velocity, we need to differentiate the equation with respect to time (t):
v = ds/dt = d(50t - 5t^2)/dt
v = 50 - 10t
Now, let's substitute the time (t) with the value for when the ball hits the sea:
v = 50 - 10(10)
v = 50 - 100
v = -50 m/s
So, the impact velocity of the ball as it enters the sea is -50 m/s. Don't worry, though, it won't be angry about its negative velocity!
c. Find the mean speed of the ball for its entire time of flight.
The mean speed can be found by calculating the total distance traveled divided by the total time taken.
First, let's find the total distance traveled. We know that the ball is projected vertically upwards from a 55 m cliff. So, the highest point it reaches is 55 m. When it falls back to the sea, the distance traveled is again 55 m. Therefore, the total distance traveled is 55 + 55 = 110 m.
Next, we need to find the total time taken. Since we have already found that the ball hits the sea after 10 seconds, the total time taken is 10 seconds.
Now, let's calculate the mean speed:
Mean Speed = Total Distance / Total Time
Mean Speed = 110 m / 10 s
Mean Speed = 11 m/s.
So, the mean speed of the ball for its entire time of flight is 11 m/s. Just remember, it's mean in a funny way, not in an angry way!
a. To find when the tennis ball hits the sea, we need to find the value of t when the height (s) becomes zero.
Given: s = 50t - 5t^2
Setting s to zero:
0 = 50t - 5t^2
This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -5, b = 50, and c = 0. Plugging the values into the formula:
t = (-50 ± √(50^2 - 4(-5)(0))) / (2(-5))
Simplifying further:
t = (-50 ± √(2500)) / (-10)
t = (-50 ± 50) / (-10)
t = 0 or t = 10
The tennis ball hits the sea either at t = 0 or t = 10 seconds.
b. To find the impact velocity of the ball as it enters the sea, we can differentiate the height equation with respect to time (t) and find the derivative.
Given: s = 50t - 5t^2
Differentiating with respect to t:
ds/dt = 50 - 10t
The impact velocity is the velocity at the moment the ball hits the sea, which is when t = 10 (from part a).
v = ds/dt = 50 - 10t
v = 50 - 10(10)
v = 50 - 100
v = -50 m/s (negative sign indicates downward)
The impact velocity of the ball as it enters the sea is -50 m/s.
c. To find the mean speed of the ball for its entire time of flight, we need to calculate the total distance traveled and divide it by the total time taken.
The total distance traveled is the difference in height between the initial position (top of the cliff) and the final position (at the sea level). In this case, it is 55 meters.
The total time taken is the time at which the ball hits the sea, which is t = 10 seconds (from part a).
Mean speed = Total distance / Total time
Mean speed = 55 / 10
Mean speed = 5.5 m/s
The mean speed of the ball for its entire time of flight is 5.5 m/s.
To solve the given problem, we need to find the value of time when the tennis ball hits the sea, then calculate its impact velocity, and finally determine the mean speed of the ball for its entire time of flight. Let's go step by step.
a. When does the tennis ball hit the sea?
To find when the tennis ball hits the sea, we need to determine the height when s equals zero. In other words, we need to solve the equation s = 50t - 5t^2 for t when s is equal to zero.
0 = 50t - 5t^2
This is a quadratic equation in the form of at^2 + bt + c = 0, where a = -5, b = 50, and c = 0. We can solve this equation by factoring or using the quadratic formula.
If we factor out t from the equation, we get:
0 = t(50 - 5t)
From this factorization, we can see that either t = 0 or 50 - 5t = 0.
If we solve 50 - 5t = 0, we find t = 10.
Therefore, the tennis ball hits the sea after 10 seconds.
b. Find the impact velocity of the ball as it enters the sea.
The impact velocity of the ball can be obtained by finding the derivative of the height function with respect to time (t) and evaluating it at the time when the ball hits the sea (t = 10 seconds).
We are given the height function s = 50t - 5t^2.
To find the velocity function, we differentiate s with respect to t:
v = ds/dt = d/dt (50t - 5t^2) = 50 - 10t.
Therefore, the velocity function is v = 50 - 10t.
To find the impact velocity, we substitute t = 10 into the velocity function:
v = 50 - 10(10) = 50 - 100 = -50 m/s.
The negative sign indicates that the velocity is directed downwards. Therefore, the impact velocity of the ball as it enters the sea is 50 m/s downward.
c. Find the mean speed of the ball for its entire time of flight.
The mean speed of an object is defined as the total distance traveled divided by the total time taken. In this case, the distance traveled by the ball is the total change in height, which is the absolute value of the initial height (55 m).
The total time taken for the ball's flight is the time it took to reach the maximum height and then descend back to sea level. Since the ball reaches the maximum height at t = 10/2 = 5 seconds, the total time taken is 5 + 5 = 10 seconds.
Therefore, the mean speed of the ball for its entire time of flight is:
Mean speed = Total distance / Total time = 55 m / 10 s = 5.5 m/s.
V = Vo + g*Tr = 0,
50 + (-10)Tr = 0,
Tr = 5 seconds. = Rise time.
S = 50*5 - 5*5^2 = 125 m. = Max ht. above bldg.
125 + 55 = 180 m. = max ht. above gnd.
h = 0.5g*Tf^2 = 180,
5Tf^2 = 180,
Tf = 6 seconds. = Fall time from max ht. to gnd.
a. T = Tr + Tf = 5 + 6 = 11 seconds.
b. V^2 = Vo^2 + 2g*h = 0 + 20*180 = 3600,
V = 60 m/s.