A tennis ball (typical mass 57.5 g) flying through the air may be affected by air resistance. The drag force applied to the ball can be approximated as: F drag == -C₂PAV² Where: Fdrag Drag force (N) Cd Coefficient of drag, which for a tennis ball is around 0.55 p = The density of air, which at ground level at 25°C is around 1.2 A = The presented area of the tennis ball in the direction of travel, around 3500 mm² v = The velocity (by convention in the positive direction) in m/s (a) Draw a free-body diagram of the tennis ball during flight, including air resistance. Also include a separate vector showing the direction of travel (v) at an arbitrary angle (theta) from horizontal. (b) Write Newton's second law for the x and y directions, and from this state the first-order differential equations (in each coordinate) that govern the motion of the tennis ball The tennis ball is at a vertical height of 1.4 m above the ground when it is struck with a racquet. The initial velocity as it leaves the racquet is horizontal at a speed of 110 km/h. Vertical height (c) Calculate how far the ball would travel before bouncing if air resistance was considered negligible. Give your answer in metres, to at least 3 significant figures. Answer:
A tennis ball (typical mass 57.5 g) flying through the air may be affected by air resistance. The drag force applied to the ball can be approximated as: F drag == -C₂PAV² Where: Fdrag Drag force (N) Cd Coefficient of drag, which for a tennis ball is around 0.55 p = The density of air, which at ground level at 25°C is around 1.2 A = The presented area of the tennis ball in the direction of travel, around 3500 mm² v = The velocity (by convention in the positive direction) in m/s (a) Draw a free-body diagram of the tennis ball during flight, including air resistance. Also include a separate vector showing the direction of travel (v) at an arbitrary angle (theta) from horizontal. (b) Write Newton's second law for the x and y directions, and from this state the first-order differential equations (in each coordinate) that govern the motion of the tennis ball The tennis ball is at a vertical height of 1.4 m above the ground when it is struck with a racquet. The initial velocity as it leaves the racquet is horizontal at a speed of 110 km/h. Vertical height (c) Calculate how far the ball would travel before bouncing if air resistance was considered negligible. Give your answer in metres, to at least 3 significant figures. Answer:
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Transcribed Image Text:A tennis ball (typical mass 57.5 g) flying through the air may be affected by air resistance. The drag force applied to the ball can be
approximated as:
F
drag
C₁pAv²
Where:
Fdrag Drag force (N)
Cd = Coefficient of drag, which for a tennis ball is around 0.55
p = The density of air, which at ground level at 25°C is around 1.2
A = The presented area of the tennis ball in the direction of travel, around 3500 mm²
v = The velocity (by convention in the positive direction) in m/s
(a) Draw a free-body diagram of the tennis ball during flight, including air resistance. Also include a separate vector showing the direction
of travel (v) at an arbitrary angle (theta) from horizontal.
(b) Write Newton's second law for the x and y directions, and from this state the first-order differential equations (in each coordinate) that
govern the motion of the tennis ball
The tennis ball is at a vertical height of 1.4 m above the ground when it is struck with a racquet. The initial velocity as it leaves the racquet
is horizontal at a speed of 110 km/h.
Vertical height
(c) Calculate how far the ball would travel before bouncing if air resistance was considered negligible. Give your answer in metres, to at
least 3 significant figures.
Answer:
We can now use our knowledge of ODEs to model the change when air resistance is significant
(d) Using Heun's method and a step size of 0.01, solve the differential equation in the x-direction to find the horizontal velocity after 0.5
seconds. Given your answer in km/h to at least 1 decimal place.
Answer:
(e) Consider the expected shape of the graph of vx over time. If you were to use Euler's method to solve for the final velocity at t = 0.5,
what predictions could you make for your calculated answer compared to the true answer? Explain your answer in 2-3 sentences. You
may wish to draw a diagram to aid your explanation.
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