When chasing a hare along a flat stretch of ground, a greyhound leaps into the air at a speed of 10.2 m/s, at an angle of 16.3° above the horizontal. (a) What is the range of his leap? (b) For how much time is he in the air?

(a) Range = (Vo^2/g)*sin(32.6)

(b) Time aloft = (2Vo/g)*sin(16.3)

To find the range of the greyhound's leap, we need to calculate the horizontal distance covered during the leap.

The horizontal component of velocity can be found using the equation:
Vx = V * cos(θ),

where Vx is the horizontal component of velocity, V is the initial velocity, and θ is the angle above the horizontal.

Plugging in the given values, we can calculate Vx:
Vx = 10.2 m/s * cos(16.3°) = 9.629 m/s.

Now, we can calculate the time the greyhound spends in the air. The total time can be found using the equation:
t = 2 * Vy / g,

where Vy is the vertical component of velocity and g is the acceleration due to gravity (approximately 9.8 m/s^2).

First, we need to calculate Vy using the equation:
Vy = V * sin(θ),

where Vy is the vertical component of velocity, V is the initial velocity, and θ is the angle above the horizontal.

Plugging in the given values, we can calculate Vy:
Vy = 10.2 m/s * sin(16.3°) = 2.825 m/s.

Now, we can calculate the time:
t = 2 * 2.825 m/s / 9.8 m/s^2 = 0.577 s.

The range of the greyhound's leap is equal to the horizontal distance traveled during this time. To find this distance, we can use the equation:
range = Vx * t,

where range is the horizontal distance and t is the time.

Plugging in the calculated values, we can find the range of the leap:
range = 9.629 m/s * 0.577 s = 5.562 m.

Therefore, the range of the greyhound's leap is approximately 5.562 meters, and it spends approximately 0.577 seconds in the air.

To find the range of the greyhound's leap, we can use the horizontal component of its velocity. The horizontal velocity can be found by multiplying the overall velocity by the cosine of the angle.

(a) Range = Horizontal Velocity x Time

The horizontal velocity (Vx) = Velocity x cos(angle)
Vx = 10.2 m/s x cos(16.3°)

The time (t) can be found using the formula:
Time = Range / Horizontal Velocity

Therefore, to calculate the range, we need to find the horizontal velocity and then divide the range by the horizontal velocity.

(b) Time = Range / Horizontal Velocity

Let's calculate the values step-by-step:

Step 1: Calculate the horizontal velocity (Vx)
Vx = 10.2 m/s x cos(16.3°)
= 10.2 m/s x 0.9599
≈ 9.80 m/s

Step 2: Calculate the range
Range = Horizontal Velocity x Time
Range = 9.80 m/s x t

Now, let's calculate time (t):

From the given information, we don't directly have the time. However, we do know the vertical component of the velocity, which is the initial vertical velocity (Vy) when the greyhound leaps.

The vertical velocity (Vy) can be found by multiplying the overall velocity by the sine of the angle.

Vy = Velocity x sin(angle)
Vy = 10.2 m/s x sin(16.3°)

We know that the greyhound will experience free-fall motion vertically, with an initial vertical velocity and acceleration due to gravity (9.8 m/s²).

Using the kinematic equation:

Vy = Vy₀ + a * t

where Vy is the final vertical velocity, Vy₀ is the initial vertical velocity, a is the acceleration, and t is the time.

At the highest point of the leap, the vertical velocity will be zero, so we can set Vy = 0 and solve for time.

0 = Vy₀ - 9.8 m/s² * t

Solving for t:

t = Vy₀ / (a)
t = Vy₀ / (9.8 m/s²)

Now we substitute the values:

t = [10.2 m/s x sin(16.3°)] / (9.8 m/s²)

After calculating, we find:

t ≈ 1.84 seconds

Then we can calculate the range:

Range = Horizontal Velocity x Time
Range = 9.80 m/s x 1.84 seconds

After calculating, we find:

(a) The range of the greyhound's leap is approximately 18 meters.
(b) The greyhound is in the air for approximately 1.84 seconds.