A shot putter throws the shot (mass=7.3kg) with an initial speed of 14.4 m/s at a 34.0 degree angle to the horizontal. Calculate the horizontal distance traveled by the shot if it leaves the athlete's hand at a height of 2.10m above the ground.
Who cares what the mass is ?
horizontal problem:
u = 14.4 cos 34 forever, calculate u
vertical problem:
Vi = 14.4 sin 34 initial speed up --- calculate Vi
h = 2.10 + Vi t - 4.9 t^2
at ground
0 = 2.10 + Vi t - 4.9 t^2
solve quadratic for t
go back to horizontal equation with that t
range = u t
To calculate the horizontal distance traveled by the shot, we will first find the time of flight and then use it to calculate the horizontal distance using the formula for horizontal displacement.
Step 1: Find the time of flight:
The vertical motion of the shot can be analyzed separately. We can use the following equations of motion to find the time of flight (t) for a projectile:
Vertical displacement (h) = (initial vertical velocity (Viy) * time (t)) - (0.5 * acceleration due to gravity (g) * time^2)
Since it leaves the athlete's hand at a height of 2.10m above the ground, the vertical displacement is -2.10 m (negative because it is moving downwards).
-2.10 m = (14.4 m/s * sin(34°) * t) - (0.5 * 9.8 m/s^2 * t^2)
Simplifying the equation, we have:
-2.10 m = 9.6 m/s * t - 4.9 m/s^2 * t^2
This is a quadratic equation, so let's solve it for t. Rearranging the equation, we get:
4.9 t^2 - 9.6 t - 2.10 = 0
Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / 2a, where
a = 4.9
b = -9.6
c = -2.10
Plugging in the values into the formula, we get:
t = (-(-9.6) ± √((-9.6)^2 - 4 * 4.9 * (-2.10))) / (2 * 4.9)
Solving this equation using a calculator, we find two possible values for t: t = 0.35s or t = 1.13s.
Since we are looking for the time it takes for the shot to reach the ground, we can discard the negative value. Therefore, the time of flight is t = 1.13s.
Step 2: Calculate the horizontal distance traveled by the shot:
Now that we have the time of flight, we can use the horizontal velocity to find the horizontal distance traveled by the shot.
The horizontal velocity of the shot (Vix) can be calculated using:
Horizontal velocity (Vix) = initial velocity (Vi) * cos(angle)
Vix = 14.4 m/s * cos(34°)
Vix ≈ 11.95 m/s
Using the formula for horizontal displacement, we have:
Horizontal distance (d) = Vix * t
d = 11.95 m/s * 1.13s
By multiplying these values together, we find:
d ≈ 13.54 meters
Therefore, the horizontal distance traveled by the shot is approximately 13.54 meters.
To calculate the horizontal distance traveled by the shot put, we can break down the initial velocity into horizontal and vertical components.
1. Horizontal component:
The horizontal component of the initial velocity can be determined using the formula:
Vx = V * cosθ
where Vx is the horizontal component of the velocity, V is the initial speed of the shot put (14.4 m/s), and θ is the launch angle (34.0 degrees).
Plug in the values:
Vx = 14.4 m/s * cos(34.0°)
Vx ≈ 11.972 m/s
2. Vertical component:
The vertical component of the initial velocity can be determined using the formula:
Vy = V * sinθ
where Vy is the vertical component of the velocity, V is the initial speed of the shot put (14.4 m/s), and θ is the launch angle (34.0 degrees).
Plug in the values:
Vy = 14.4 m/s * sin(34.0°)
Vy ≈ 8.085 m/s
3. Time of flight:
We can find the time of flight by using the vertical displacement and the vertical component of velocity.
Δy = Vy * t + (1/2) * g * t^2
where Δy is the vertical displacement (2.10 m), Vy is the vertical component of the velocity (8.085 m/s), t is the time, and g is the acceleration due to gravity (9.8 m/s^2).
Rearrange the equation:
2.10 m = 8.085 m/s * t + (1/2) * 9.8 m/s^2 * t^2
Rearrange further:
4.9 m/s^2 * t^2 + 8.085 m/s * t - 2.10 m = 0
Solve this quadratic equation to find the positive value of t.
4. Horizontal distance:
The horizontal distance traveled by the shot can be calculated using the time of flight and the horizontal component of velocity.
Dx = Vx * t
where Dx is the horizontal distance, Vx is the horizontal component of the velocity (11.972 m/s), and t is the time of flight you found in step 3.
Plug in the values:
Dx = 11.972 m/s * t
By following these calculations, you can find the horizontal distance traveled by the shot put.